A Variational Analysis of a Gauged Nonlinear Schrödinger Equation

Size: px
Start display at page:

Download "A Variational Analysis of a Gauged Nonlinear Schrödinger Equation"

Transcription

1 A Variational Analysis of a Gauged Nonlinear Schrödinger Equation Alessio Pomponio, joint work with David Ruiz Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Variational and Topological Methods in Nonlinear Phenomena Alghero, June 24-28, 2013.

2 We are concerned with a planar gauged Nonlinear Schrödinger Equation: id 0 φ + (D 1 D 1 + D 2 D 2 )φ + φ p 1 φ = 0. Here t R, x = (x 1, x 2 ) R 2, φ : R R 2 C is the scalar field, A µ : R R 2 R are the components of the gauge potential, namely (A 0, A 1, A 2 ) = (A 0, A), and D µ = µ + ia µ is the covariant derivative (µ = 0, 1, 2).

3 The field equations, in non-relativistic notation, are B = A, E = A 0 t A, where (E, B) is the electromagnetic field. The natural and obvious equation of gauge field dynamic is the Maxwell equation: µ F µν = j ν, where F µν = µ A ν ν A µ, and j µ is the conserved matter current, j 0 = φ 2, j i = 2Im ( φd i φ).

4 A modified gauge field equation has been introduced adding the so-called Chern-Simons term into the previous equation [Deser, Hangen, Jackiw, Schonfeld, Templeton, in the 80s]:

5 A modified gauge field equation has been introduced adding the so-called Chern-Simons term into the previous equation [Deser, Hangen, Jackiw, Schonfeld, Templeton, in the 80s]: µ F µν κɛναβ F αβ = j ν, where κ is a parameter that measures the strength of the Chern-Simons term, ɛ ναβ is the Levi-Civita tensor, and super-indices are related to the Minkowski metric with signature (1, 1, 1).

6 A modified gauge field equation has been introduced adding the so-called Chern-Simons term into the previous equation [Deser, Hangen, Jackiw, Schonfeld, Templeton, in the 80s]: µ F µν κɛναβ F αβ = j ν, where κ is a parameter that measures the strength of the Chern-Simons term, ɛ ναβ is the Levi-Civita tensor, and super-indices are related to the Minkowski metric with signature (1, 1, 1). At low energies, the Maxwell term becomes negligible and can be dropped, giving rise to: 1 2 κɛναβ F αβ = j ν.

7 Taking for simplicity κ = 2, we arrive to the system id 0 φ + (D 1 D 1 + D 2 D 2 )φ + φ p 1 φ = 0, 0 A 1 1 A 0 = Im( φd 2 φ), 0 A 2 2 A 0 = Im( φd 1 φ), 1 A 2 2 A 1 = 1 2 φ 2, (1) where the unknowns are (φ, A 0, A 1, A 2 ).

8 Taking for simplicity κ = 2, we arrive to the system id 0 φ + (D 1 D 1 + D 2 D 2 )φ + φ p 1 φ = 0, 0 A 1 1 A 0 = Im( φd 2 φ), 0 A 2 2 A 0 = Im( φd 1 φ), 1 A 2 2 A 1 = 1 2 φ 2, (1) where the unknowns are (φ, A 0, A 1, A 2 ). As usual in Chern-Simons theory, problem (1) is invariant under gauge transformation, for any arbitrary C -function χ. φ φe iχ, A µ A µ µ χ,

9 This model was first proposed and studied by Jackiw & Pi, in several papers in the beginning of 90s, and sometimes has received the name of Chern-Simons-Schrödinger equation.

10 This model was first proposed and studied by Jackiw & Pi, in several papers in the beginning of 90s, and sometimes has received the name of Chern-Simons-Schrödinger equation. The initial value problem, as well as global existence and blow-up, has been addressed in [Bergé, de Bouard & Saut, 1995; Huh, ] for the case p = 3.

11 The existence of stationary states for (1) and general p > 1 has been studied recently in [Byeon, Huh & Seok, JFA 2012].

12 The existence of stationary states for (1) and general p > 1 has been studied recently in [Byeon, Huh & Seok, JFA 2012]. They look for the classical solutions, that is, solutions (φ, A 0, A 1, A 2 ) of (1) in C 2 (R 2 ) C 1 (R 2 ) C 1 (R 2 ) C 1 (R 2 ).

13 The existence of stationary states for (1) and general p > 1 has been studied recently in [Byeon, Huh & Seok, JFA 2012]. They look for the classical solutions, that is, solutions (φ, A 0, A 1, A 2 ) of (1) in C 2 (R 2 ) C 1 (R 2 ) C 1 (R 2 ) C 1 (R 2 ). Moreover they seek the standing wave solutions of the form φ(t, x) = u( x )e iωt, A 0 (x) = k( x ), A 1 (t, x) = x 2 x 2 h( x ), A 2(t, x) = x 1 x 2 h( x ), where ω > 0 is a given frequency and u, k, h are real valued functions on [0, ) and h(0) = 0.

14 With this ansatz, (1) becomes u + ωu + A 0 u + A 2 1 u + A2 2 u u p 1 u = 0, 1 A 0 = A 2 u 2, 2 A 0 = A 1 u 2, 1 A 2 2 A 1 = 1 2 u2.

15 With this ansatz, (1) becomes u + ωu + A 0 u + A 2 1 u + A2 2 u u p 1 u = 0, 1 A 0 = A 2 u 2, 2 A 0 = A 1 u 2, 1 A 2 2 A 1 = 1 2 u2. First, they solve A 0, A 1, A 2 in terms of u. The ansatz implies 1 s h (s) = 1 2 u2 (s).

16 With this ansatz, (1) becomes u + ωu + A 0 u + A 2 1 u + A2 2 u u p 1 u = 0, 1 A 0 = A 2 u 2, 2 A 0 = A 1 u 2, 1 A 2 2 A 1 = 1 2 u2. First, they solve A 0, A 1, A 2 in terms of u. The ansatz implies 1 s h (s) = 1 2 u2 (s). From the condition h(0) = 0, we get and so h(r) = r 0 s 2 u2 (s) ds, A 1 (x) = x x 2 s x u2 (s) ds, A 2 (x) = x x 1 s x u2 (s) ds.

17 For what concerns A 0, we have k (s) = h(s) u 2 (s). s By integrating both sides from r to, we obtain A 0 (x) = k( x ) = ξ + x h(s) u 2 (s) ds, s where ξ is an arbitrary constant which is just the value of k at infinity.

18 For what concerns A 0, we have k (s) = h(s) u 2 (s). s By integrating both sides from r to, we obtain A 0 (x) = k( x ) = ξ + x h(s) u 2 (s) ds, s where ξ is an arbitrary constant which is just the value of k at infinity. If u H 1 r (R 2 ) C(R 2 ), then A 0, A 1, A 2 L (R 2 ) C 1 (R 2 ).

19 Therefore we need only to solve, in R 2, the equation: ( u + ω + ξ + h2 ( x ) + x 2 + x ) h(s) u 2 (s) ds u = u p 1 u. s

20 Therefore we need only to solve, in R 2, the equation: ( u + ω + ξ + h2 ( x ) + x 2 + x ) h(s) u 2 (s) ds u = u p 1 u. s Let us show that the constant ω + ξ is a gauge invariant of the stationary solutions of the problem.

21 Since problem (1) is invariant under gauge transformation, φ φe iχ, A µ A µ µ χ, for any arbitrary C function χ, if ( (φ, A 0, A 1, A 2 ) = u( x )e iωt, k( x ), x ) 2 x 2 h( x ), x 1 x 2 h( x ), is a solution of (1),

22 Since problem (1) is invariant under gauge transformation, φ φe iχ, A µ A µ µ χ, for any arbitrary C function χ, if ( (φ, A 0, A 1, A 2 ) = u( x )e iωt, k( x ), x ) 2 x 2 h( x ), x 1 x 2 h( x ), is a solution of (1), then, taking χ = ct, ( ( φ, Ã 0, A 1, A 2 ) = u( x )e i(ω+c)t, k( x ) c, x ) 2 x 2 h( x ), x 1 x 2 h( x ), is a solution of (1), too.

23 The equation We will take ξ = 0 in what follows, that is, lim A 0 (x) = 0, x + which was assumed in [Jackiw & Pi, Bergé, de Bouard & Saut].

24 The equation We will take ξ = 0 in what follows, that is, lim A 0 (x) = 0, x + which was assumed in [Jackiw & Pi, Bergé, de Bouard & Saut]. Our aim is to solve, in R 2, the nonlocal equation: ( u + ω + h2 ( x ) + x 2 + x ) h(s) u 2 (s) ds u = u p 1 u, s (P) where h(r) = r 0 s 2 u2 (s) ds.

25 In [Byeon, Huh, Seok, JFA 2012] it is shown that (P) is indeed the Euler-Lagrange equation of the energy functional: I ω : H 1 r (R 2 ) R, defined as I ω (u) = 1 ( u 2 + ωu 2) dx 2 R u 2 ( (x) x ) 2 8 R 2 x 2 su 2 (s) ds dx 1 u p+1 dx. 0 p + 1 R 2

26 In [Byeon, Huh, Seok, JFA 2012] it is shown that (P) is indeed the Euler-Lagrange equation of the energy functional: I ω : H 1 r (R 2 ) R, defined as I ω (u) = 1 ( u 2 + ωu 2) dx 2 R u 2 ( (x) x ) 2 8 R 2 x 2 su 2 (s) ds dx 1 u p+1 dx. 0 p + 1 R 2

27 Since ( 1 x ) 2 x 2 su 2 (s) ds = C ( ) 2 0 x 2 u 2 (x) dx C u 4 L B(0, x ) 4 (R 2 ),

28 Since ( 1 x ) 2 x 2 su 2 (s) ds = C ( ) 2 0 x 2 u 2 (x) dx C u 4 L B(0, x ) 4 (R 2 ), then, for any u H 1 r (R 2 ), u 2 (x) R 2 x 2 ( x 2 su 2 (s) ds) dx < +. 0

29 Since ( 1 x ) 2 x 2 su 2 (s) ds = C ( ) 2 0 x 2 u 2 (x) dx C u 4 L B(0, x ) 4 (R 2 ), then, for any u H 1 r (R 2 ), u 2 (x) R 2 x 2 ( x 2 su 2 (s) ds) dx < +. 0 Therefore the functional is well defined in H 1 r (R 2 ).

30 A useful inequality In [Byeon, Huh & Seok], it is proved that, for any u H 1 r (R 2 ), u(x) 4 dx R 2 ( ) 1 ( 2 u(x) 2 2 u 2 ( x 2 12 dx R 2 R 2 x 2 su (s)ds) dx) 2. 0

31 A useful inequality In [Byeon, Huh & Seok], it is proved that, for any u H 1 r (R 2 ), u(x) 4 dx R 2 ( ) 1 ( 2 u(x) 2 2 u 2 ( x 2 12 dx R 2 R 2 x 2 su (s)ds) dx) 2. 0 Furthermore, the equality is attained by a continuum of functions { } 8l u l = 1 + lx 2 H1 r (R 2 ) : l (0, + ).

32 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term.

33 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below.

34 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountain-pass geometry.

35 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountain-pass geometry. The existence of a solution is not so direct, since for p (3, 5) the (PS) property is not known to hold: we do not know if (PS)-sequences are bounded.

36 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountain-pass geometry. The existence of a solution is not so direct, since for p (3, 5) the (PS) property is not known to hold: we do not know if (PS)-sequences are bounded. This problem is bypassed by using a constrained minimization taking into account the Nehari and Pohozaev identities, in the spirit of [Ruiz, JFA 2006] for the Schrödinger-Poisson equation.

37 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountain-pass geometry. The existence of a solution is not so direct, since for p (3, 5) the (PS) property is not known to hold: we do not know if (PS)-sequences are bounded. This problem is bypassed by using a constrained minimization taking into account the Nehari and Pohozaev identities, in the spirit of [Ruiz, JFA 2006] for the Schrödinger-Poisson equation. Infinitely many solutions have been found in [Huh, JMP 2012] for p > 5 (possibly sign-changing): this case is more easy since (PS)-condition holds.

38 Byeon-Huh-Seok results: case p = 3 This is a special case:

39 Byeon-Huh-Seok results: case p = 3 This is a special case: static solutions can be found by passing to a self-dual equation, which leads to a Liouville equation that can be solved explicitly.

40 Byeon-Huh-Seok results: case p = 3 This is a special case: static solutions can be found by passing to a self-dual equation, which leads to a Liouville equation that can be solved explicitly. Any standing wave solutions (φ, A 0, A 1, A 2 ) of the previous type with φ > 0 have the following form: (φ, A 0, A 1, A 2 ) = ( ( 8le iωt 1 + lx 2, 2l 1 + lx 2 where l > 0 is an arbitrary real constant. ) 2 ω, 2l 2 x lx 2, 2l 2 x lx 2 ),

41 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity.

42 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. Solutions are found as minimizers on a L 2 -sphere.

43 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. Solutions are found as minimizers on a L 2 -sphere. The value ω comes out as a Lagrange multiplier, and it is not controlled.

44 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. Solutions are found as minimizers on a L 2 -sphere. The value ω comes out as a Lagrange multiplier, and it is not controlled. By the gauge invariance, this is not a problem if we are looking for solutions (φ, A 0, A 1, A 2 ) of the entire system (1).

45 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. Solutions are found as minimizers on a L 2 -sphere. The value ω comes out as a Lagrange multiplier, and it is not controlled. By the gauge invariance, this is not a problem if we are looking for solutions (φ, A 0, A 1, A 2 ) of the entire system (1). For what concerns the single equation (P), a solution u is found only for a particular value of ω.

46 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. Solutions are found as minimizers on a L 2 -sphere. The value ω comes out as a Lagrange multiplier, and it is not controlled. By the gauge invariance, this is not a problem if we are looking for solutions (φ, A 0, A 1, A 2 ) of the entire system (1). For what concerns the single equation (P), a solution u is found only for a particular value of ω. The global behavior of the energy functional I ω is not studied.

47 Analogy with Schrödinger-Poisson equation?

48 Analogy with Schrödinger-Poisson equation? The nonlinear Schrödinger-Poisson equation is { u + u + λφu = u p 1 u in R 3, φ = u 2 in R 3, (2) here λ > 0. If ū is a critical point of the functional and if I λ (u) = 1 ( u 2 + u 2 )dx + λ 2 R u p+1 dx, p + 1 R 3 then (ū, φū) is a solution of (2). φū = 1 x ū 2, R 3 ( ) 1 x u 2 u 2 dx

49 Analogy with Schrödinger-Poisson equation? First difference: while the critical exponent for problem (P) is p = 3, for (2) it is p = 2.

50 Analogy with Schrödinger-Poisson equation? First difference: while the critical exponent for problem (P) is p = 3, for (2) it is p = 2. The case 1 < p < 3 for problem (P) is similar to the case 1 < p < 2 for (2)?

51 Analogy with Schrödinger-Poisson equation? First difference: while the critical exponent for problem (P) is p = 3, for (2) it is p = 2. The case 1 < p < 3 for problem (P) is similar to the case 1 < p < 2 for (2)? In particular, in [Ruiz, JFA 2006] it is proved that if 1 < p < 2, for λ small enough, the functional I λ is bounded from below and it possesses at least two critical points: one is the minimum, at negative level, the other is a mountain-pass critical point, at positive level.

52 On the boundedness from below of Iω

53 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). There exists ω 0 such that: if ω (0, ω 0 ), then I ω is unbounded from below;

54 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). There exists ω 0 such that: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0;

55 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). There exists ω 0 such that: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive.

56 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). There exists ω 0 such that: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive. ω 0 has an explicit expression: ω 0 = 3 p 3 + p 3 p 1 2(3 p) p ( m 2 (3 + p) p 1 ) p 1 2(3 p), with m = + ( ( )) 2 2 p 1 p + 1 cosh2 2 r 1 p dr.

57 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain.

58 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain. There exists a minimizer u n on the ball B(0, n) with Dirichlet boundary conditions.

59 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain. There exists a minimizer u n on the ball B(0, n) with Dirichlet boundary conditions. To prove boundedness of u n, the problem is the possible loss of mass at infinity as n +. We need to study the behavior of those masses.

60 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain. There exists a minimizer u n on the ball B(0, n) with Dirichlet boundary conditions. To prove boundedness of u n, the problem is the possible loss of mass at infinity as n +. We need to study the behavior of those masses. If unbounded, the sequence u n behaves as a soliton, if u n is interpreted as a function of a single real variable.

61 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain. There exists a minimizer u n on the ball B(0, n) with Dirichlet boundary conditions. To prove boundedness of u n, the problem is the possible loss of mass at infinity as n +. We need to study the behavior of those masses. If unbounded, the sequence u n behaves as a soliton, if u n is interpreted as a function of a single real variable. I ω admits a natural approximation through a limit functional.

62 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain. There exists a minimizer u n on the ball B(0, n) with Dirichlet boundary conditions. To prove boundedness of u n, the problem is the possible loss of mass at infinity as n +. We need to study the behavior of those masses. If unbounded, the sequence u n behaves as a soliton, if u n is interpreted as a function of a single real variable. I ω admits a natural approximation through a limit functional. The critical points of that limit functional, and their energy, can be found explicitly, so we can find ω 0.

63 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +.

64 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. (2π) 1 I ω (u ρ ) = p + 1 ( u ρ 2 + ωu 2 ρ)r dr u 2 ( ρ(r) r 2 su 2 r ρ(s) ds) dr u ρ p+1 r dr.

65 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. (2π) 1 I ω (u ρ ) = ρ + 1 p + 1 ( u 2 + ωu 2 )(r + ρ) dr ρ + u 2 ( (r) r r + ρ ρ 2 (s + ρ)u 2 (s) ds) dr ρ u p+1 (r + ρ) dr.

66 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. (2π) 1 I ω (u ρ ) p + 1 ( u 2 + ωu 2 )(r + ρ) dr + u 2 ( (r) r r + ρ 2 (s + ρ)u 2 (s) ds) dr u p+1 (r + ρ) dr.

67 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. (2π) 1 I ω (u ρ ) p + 1 ( u 2 + ωu 2 )(r + ρ) dr + u 2 ( (r) r r + ρ u p+1 (r + ρ) dr. (s + ρ)u 2 (s) ds) 2 dr

68 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. (2π) 1 I ω (u ρ ) p + 1 ( u 2 + ωu 2 )ρ dr + u 2 ( (r) r ρ u p+1 ρ dr. ρu 2 (s) ds) 2 dr

69 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. [ 1 + (2π) 1 I ω (u ρ ) ρ p + 1 ( u 2 + ωu 2 ) dr + ( r u 2 (r) 2 u 2 (s) ds) dr ] u p+1 dr.

70 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. [ 1 + (2π) 1 I ω (u ρ ) ρ p + 1 ( u 2 + ωu 2 ) dr + ( r u 2 (r) 2 u 2 (s) ds) dr ] u p+1 dr.

71 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. [ 1 + (2π) 1 I ω (u ρ ) ρ ( u 2 + ωu 2 ) dr ( + 1 p ) 3 u 2 (r)dr ] u p+1 dr.

72 The limit functional Let u a fixed function, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +. [ 1 + (2π) 1 I ω (u ρ ) ρ ( u 2 + ωu 2 ) dr ( + 1 p ) 3 u 2 (r)dr ] u p+1 dr.

73 It is natural to define the limit functional J ω : H 1 (R) R, J ω (u) = p + 1 ( u 2 + ωu 2) dr + 1 ( u p+1 dr. ) 3 u 2 dr

74 It is natural to define the limit functional J ω : H 1 (R) R, We have J ω (u) = p + 1 ( u 2 + ωu 2) dr + 1 ( u p+1 dr. ) 3 u 2 dr I ω (u ρ ) 2πρ J ω (u), as ρ +.

75 It is natural to define the limit functional J ω : H 1 (R) R, We have J ω (u) = p + 1 ( u 2 + ωu 2) dr + 1 ( u p+1 dr. ) 3 u 2 dr I ω (u ρ ) 2πρ J ω (u), as ρ +. Of course inf J ω < 0 inf I ω =.

76 It is natural to define the limit functional J ω : H 1 (R) R, We have J ω (u) = p + 1 ( u 2 + ωu 2) dr + 1 ( u p+1 dr. ) 3 u 2 dr I ω (u ρ ) 2πρ J ω (u), as ρ +. Of course We will show inf J ω < 0 inf I ω =. inf J ω < 0 inf I ω =.

77 The limit functional Proposition Let p (1, 3) and ω > 0. Then: a) J ω is coercive and attains its infimum;

78 The limit functional Proposition Let p (1, 3) and ω > 0. Then: a) J ω is coercive and attains its infimum; b) 0 is a local minimum of J ω ;

79 The limit functional Proposition Let p (1, 3) and ω > 0. Then: a) J ω is coercive and attains its infimum; b) 0 is a local minimum of J ω ; c) there exists ω 0 > 0 such that min J ω < 0 if and only if ω (0, ω 0 ).

80 The limit problem The critical points of J ω are solutions of the nonlocal equation u + ωu + 1 ( + 2 u 2 (s) ds) u = u p 1 u, in R. (P ) 4

81 The limit problem The critical points of J ω are solutions of the nonlocal equation u + ωu + 1 ( + 2 u 2 (s) ds) u = u p 1 u, 4 in R. (P ) We can find the explicit solutions of problem (P ).

82 The limit problem The critical points of J ω are solutions of the nonlocal equation u + ωu + 1 ( + 2 u 2 (s) ds) u = u p 1 u, in R. (P ) 4 We can find the explicit solutions of problem (P ). Observe that (P ) can be written as [ u + ω + 1 ( + ) ] 2 u 2 (s) ds u = u p 1 u, in R. 4

83 The limit problem The critical points of J ω are solutions of the nonlocal equation u + ωu + 1 ( + 2 u 2 (s) ds) u = u p 1 u, in R. (P ) 4 We can find the explicit solutions of problem (P ). Observe that (P ) can be written as [ u + ω + 1 ( + ) ] 2 u 2 (s) ds u = u p 1 u, in R. 4 }{{} k

84 One dimensional nonlinear Schrödinger equation For any k > 0, let w k H 1 (R) be the unique positive radial solution of: w k + kw k = w p k, in R. (3)

85 One dimensional nonlinear Schrödinger equation For any k > 0, let w k H 1 (R) be the unique positive radial solution of: w k + kw k = w p k, in R. (3) Any solution of (3) is of the form u(r) = ±w k (r ξ), for some ξ R.

86 One dimensional nonlinear Schrödinger equation For any k > 0, let w k H 1 (R) be the unique positive radial solution of: w k + kw k = w p k, in R. (3) Any solution of (3) is of the form u(r) = ±w k (r ξ), for some ξ R. Moreover, w k (r) = k 1 p 1 w 1 ( kr), where w 1 (r) = ( ( )) 1 2 p 1 p + 1 cosh2 2 r 1 p.

87 One dimensional nonlinear Schrödinger equation For any k > 0, let w k H 1 (R) be the unique positive radial solution of: w k + kw k = w p k, in R. (3) Any solution of (3) is of the form u(r) = ±w k (r ξ), for some ξ R. Moreover, where w 1 (r) = In what follows we define w k (r) = k 1 p 1 w 1 ( kr), ( ( )) 1 2 p 1 p + 1 cosh2 2 r 1 p. m = + w 2 1 dr.

88 Proposition u is a nontrivial solution of (P ) if and only if u(r) = w k (r ξ) for some ξ R and k a root of k = ω m2 k 5 p p 1, k > 0. (4)

89 Proposition u is a nontrivial solution of (P ) if and only if u(r) = w k (r ξ) for some ξ R and k a root of k = ω m2 k 5 p p 1, k > 0. (4) Moreover, there exists ω 1 > ω 0 > 0 such that if ω > ω 1, (4) has no solution and there is no nontrivial solution of (P );

90 Proposition u is a nontrivial solution of (P ) if and only if u(r) = w k (r ξ) for some ξ R and k a root of k = ω m2 k 5 p p 1, k > 0. (4) Moreover, there exists ω 1 > ω 0 > 0 such that if ω > ω 1, (4) has no solution and there is no nontrivial solution of (P ); if ω = ω 1, (4) has only one solution k 0 and w k0 (r) is the only nontrivial solution of (P );

91 Proposition u is a nontrivial solution of (P ) if and only if u(r) = w k (r ξ) for some ξ R and k a root of k = ω m2 k 5 p p 1, k > 0. (4) Moreover, there exists ω 1 > ω 0 > 0 such that if ω > ω 1, (4) has no solution and there is no nontrivial solution of (P ); if ω = ω 1, (4) has only one solution k 0 and w k0 (r) is the only nontrivial solution of (P ); if ω (0, ω 1 ), (4) has two solutions k 1 (ω) < k 2 (ω) and w k1 (r), w k2 (r) are the only two nontrivial solutions of (P ).

92 Proposition u is a nontrivial solution of (P ) if and only if u(r) = w k (r ξ) for some ξ R and k a root of k = ω m2 k 5 p p 1, k > 0. (4) Moreover, there exists ω 1 > ω 0 > 0 such that if ω > ω 1, (4) has no solution and there is no nontrivial solution of (P ); if ω = ω 1, (4) has only one solution k 0 and w k0 (r) is the only nontrivial solution of (P ); if ω (0, ω 1 ), (4) has two solutions k 1 (ω) < k 2 (ω) and w k1 (r), w k2 (r) are the only two nontrivial solutions of (P ). Moreover ( ) (5 p)m 2 p 1 2(3 p) m ω 1 = 2 4(p 1) 4 ( (5 p)m 2 4(p 1) ) (5 p) 2(3 p).

93 The map ψ Let us evaluate J ω on the curve k w k. We have [ p 5 ψ(k) = J ω (w k ) = m 2(3 + p) k 3+p 2(p 1) + ω ] 2 k 5 p 2(p 1) + m2 24 k 3(5 p) 2(p 1).

94 The map ψ Let us evaluate J ω on the curve k w k. We have [ p 5 ψ(k) = J ω (w k ) = m 2(3 + p) k 3+p 2(p 1) + ω ] 2 k 5 p 2(p 1) + m2 24 k 3(5 p) 2(p 1). Then: [ d 7 3p 5 p ψ(k) = m k 2(p 1) k + ω + 1 ] dk 4(p 1) 4 m2 k 5 p p 1.

95 The map ψ Let us evaluate J ω on the curve k w k. We have [ p 5 ψ(k) = J ω (w k ) = m 2(3 + p) k 3+p 2(p 1) + ω ] 2 k 5 p 2(p 1) + m2 24 k 3(5 p) 2(p 1). Then: [ d 7 3p 5 p ψ(k) = m k 2(p 1) k + ω + 1 ] dk 4(p 1) 4 m2 k 5 p p 1. The roots of (4) are exactly the critical points of ψ.

96 The map ψ Let us evaluate J ω on the curve k w k. We have [ p 5 ψ(k) = J ω (w k ) = m 2(3 + p) k 3+p 2(p 1) + ω ] 2 k 5 p 2(p 1) + m2 24 k 3(5 p) 2(p 1). Then: [ d 7 3p 5 p ψ(k) = m k 2(p 1) k + ω + 1 ] dk 4(p 1) 4 m2 k 5 p p 1. The roots of (4) are exactly the critical points of ψ. Since 5 p 2(p 1) < 3 + p 3(5 p) < 2(p 1) 2(p 1), ψ is increasing near 0 (for ω > 0) and near infinity.

97 The map ψ If ω > ω 1, ψ is positive and increasing without critical points.

98 The map ψ If ω > ω 1, ψ is positive and increasing without critical points. If ω = ω 1, ψ is still positive and increasing, but it has an inflection point at k = k 0.

99 The map ψ If ω > ω 1, ψ is positive and increasing without critical points. If ω = ω 1, ψ is still positive and increasing, but it has an inflection point at k = k 0. If ω (ω 0, ω 1 ), ψ has a local maximum and minimum attained at k 1 and k 2, respectively, and w k2 is a local minimizer of J ω with J ω (w k2 ) > 0.

100 The map ψ If ω > ω 1, ψ is positive and increasing without critical points. If ω = ω 1, ψ is still positive and increasing, but it has an inflection point at k = k 0. If ω (ω 0, ω 1 ), ψ has a local maximum and minimum attained at k 1 and k 2, respectively, and w k2 is a local minimizer of J ω with J ω (w k2 ) > 0. If ω = ω 0, ψ(k 2 ) = 0. Observe then, in this case, the minimum of J ω0 is 0, and is attained at 0 and w k2.

101 The map ψ If ω > ω 1, ψ is positive and increasing without critical points. If ω = ω 1, ψ is still positive and increasing, but it has an inflection point at k = k 0. If ω (ω 0, ω 1 ), ψ has a local maximum and minimum attained at k 1 and k 2, respectively, and w k2 is a local minimizer of J ω with J ω (w k2 ) > 0. If ω = ω 0, ψ(k 2 ) = 0. Observe then, in this case, the minimum of J ω0 is 0, and is attained at 0 and w k2. If ω (0, ω 0 ), ψ(k 2 ) < 0 and then w k2 is the unique global minimizer of J ω and J ω (w k2 ) < 0.

102 The threshold value ω 0 In order to get the value of ω 0, observe that J ω0 (w k2 ) = 0.

103 The threshold value ω 0 In order to get the value of ω 0, observe that J ω0 (w k2 ) = 0. Therefore, ω 0 > 0 solves: and we infer that d ψ(k) = 0, dk ψ(k) = 0. ω 0 = 3 p 3 + p 3 p 1 2(3 p) p ( m 2 (3 + p) p 1 ) p 1 2(3 p).

104 The values ω 0 (p) < ω 1 (p), for p (1, 3). We cannot obtain a more explicit expression of m depending on p.

105 The values ω 0 (p) < ω 1 (p), for p (1, 3). We cannot obtain a more explicit expression of m depending on p. For some specific values of p, m can be explicitly computed, and hence ω 0 and ω 1. For instance, if p = 2, m = 6, ω 1 = and ω 0 =

106 The values ω 0 (p) < ω 1 (p), for p (1, 3). We cannot obtain a more explicit expression of m depending on p. For some specific values of p, m can be explicitly computed, and hence ω 0 and ω 1. For instance, if p = 2, m = 6, ω 1 = 2 9 and 3 ω 0 = 2 5. More in general, we have

107 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). We have: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive.

108 On the boundedness from below of I ω Theorem (A.P. & D. Ruiz) Let p (1, 3). We have: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive. Since J ω (w k2 ) < 0, if ω (0, ω 0 ), and I ω (w k2 ( ρ)) 2πρ J ω (w k2 ), as ρ +, the first part is proved.

109 Sketch of the proof: case ω > ω0

110 Sketch of the proof: case ω ω 0 I ω is coercive when the problem is posed on a bounded domain.

111 Sketch of the proof: case ω ω 0 I ω is coercive when the problem is posed on a bounded domain. There exists u n a minimizer for I ω H 1 0,r (B(0,n)). Moreover, I ω (u n ) inf I ω, as n +.

112 Sketch of the proof: case ω ω 0 I ω is coercive when the problem is posed on a bounded domain. There exists u n a minimizer for I ω H 1 0,r (B(0,n)). Moreover, I ω (u n ) inf I ω, as n +. If u n is bounded, then I ω (u n ) is also bounded and therefore inf I ω is finite. In what follows we assume that u n is an unbounded sequence.

113 Sketch of the proof: case ω ω 0 I ω is coercive when the problem is posed on a bounded domain. There exists u n a minimizer for I ω H 1 0,r (B(0,n)). Moreover, I ω (u n ) inf I ω, as n +. If u n is bounded, then I ω (u n ) is also bounded and therefore inf I ω is finite. In what follows we assume that u n is an unbounded sequence. If u n is interpreted as a function of a single real variable, u n does not vanish.

114 Sketch of the proof: case ω ω 0 Let ξ n R be the largest center of mass of u n : we have that ξ n u n 2.

115 Sketch of the proof: case ω ω 0 Let ξ n R be the largest center of mass of u n : we have that ξ n u n 2. Define ψ n : [0, + ] [0, 1] be a smooth function such that ψ n (r) = { 0, if r ξn 3 u n, 1, if r ξ n + 2 u n.

116 Sketch of the proof: case ω ω 0 Let ξ n R be the largest center of mass of u n : we have that ξ n u n 2. Define ψ n : [0, + ] [0, 1] be a smooth function such that ψ n (r) = { 0, if r ξn 3 u n, 1, if r ξ n + 2 u n. Let s estimate I ω (u n ) with I ω (ψ n u n ) and I ω ((1 ψ n )u n ): I ω (u n ) I ω (u n ψ n ) + I ω (u n (1 ψ n )) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O( u n ).

117 Sketch of the proof: conclusion for ω > ω 0 Since I ω (u n ψ n ) = 2πξ n J ω (u n ψ n ) + O( u n ).

118 Sketch of the proof: conclusion for ω > ω 0 Since we have I ω (u n ψ n ) = 2πξ n J ω (u n ψ n ) + O( u n ). I ω (u n ) 2πξ n J ω (u n ψ n ) + I ω (u n (1 ψ n )) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O( u n ).

119 Sketch of the proof: conclusion for ω > ω 0 Since we have I ω (u n ψ n ) = 2πξ n J ω (u n ψ n ) + O( u n ). I ω (u n ) 2πξ n J ω (u n ψ n ) + I ω (u n (1 ψ n )) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O( u n ). Since u n ψ n H1 (R) c, we can prove that J ω (u n ψ n ) c > 0.

120 Sketch of the proof: conclusion for ω > ω 0 Since we have I ω (u n ψ n ) = 2πξ n J ω (u n ψ n ) + O( u n ). I ω (u n ) 2πξ n J ω (u n ψ n ) + I ω (u n (1 ψ n )) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O( u n ). Since u n ψ n H1 (R) c, we can prove that J ω (u n ψ n ) c > 0. Since ξ n u n 2, we infer that I ω (u n ) > I ω (u n (1 ψ n )).

121 Sketch of the proof: conclusion for ω > ω 0 Since I ω (u n ψ n ) = 2πξ n J ω (u n ψ n ) + O( u n ). we have I ω (u n ) 2πξ n J ω (u n ψ n ) + I ω (u n (1 ψ n )) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O( u n ). Since u n ψ n H1 (R) c, we can prove that J ω (u n ψ n ) c > 0. Since ξ n u n 2, we infer that I ω (u n ) > I ω (u n (1 ψ n )). Contradiction with the definition of u n, which proves that inf I ω >.

122 Sketch of the proof: conclusion for ω > ω 0 Let us now show that I ω is coercive.

123 Sketch of the proof: conclusion for ω > ω 0 Let us now show that I ω is coercive. Take u n H 1 r (R 2 ) an unbounded sequence, and assume that I ω (u n ) is bounded from above.

124 Sketch of the proof: conclusion for ω > ω 0 Let us now show that I ω is coercive. Take u n H 1 r (R 2 ) an unbounded sequence, and assume that I ω (u n ) is bounded from above. u n L2 (R 2 ) +.

125 Sketch of the proof: conclusion for ω > ω 0 Let us now show that I ω is coercive. Take u n H 1 r (R 2 ) an unbounded sequence, and assume that I ω (u n ) is bounded from above. u n L2 (R 2 ) +. Then for any ω 0 < ω < ω, I ω (u n ) = I ω (u n ) + ω ω u n 2 2 L 2 (R 2 ), a contradiction, since we know that I ω is bounded from below.

126 Sketch of the proof: conclusion for ω = ω0

127 Sketch of the proof: conclusion for ω = ω 0 We reach a contradiction unless J ω0 (u n ψ n ) 0.

128 Sketch of the proof: conclusion for ω = ω 0 We reach a contradiction unless J ω0 (u n ψ n ) 0. This implies that ψ n u n ( ξ n ) w k2, where w k2 nontrivial minimum of J ω0. is the

129 Sketch of the proof: conclusion for ω = ω 0 We reach a contradiction unless J ω0 (u n ψ n ) 0. This implies that ψ n u n ( ξ n ) w k2, where w k2 is the nontrivial minimum of J ω0. With this extra information, we have a better estimate: I ω0 (u n ) 2πξ n J ω0 (u n ψ n ) + I ω0 ( un (1 ψ n ) ) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O(1).

130 Sketch of the proof: conclusion for ω = ω 0 We reach a contradiction unless J ω0 (u n ψ n ) 0. This implies that ψ n u n ( ξ n ) w k2, where w k2 is the nontrivial minimum of J ω0. With this extra information, we have a better estimate: I ω0 (u n ) 2πξ n J ω0 (u n ψ n ) + I ω0 ( un (1 ψ n ) ) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O(1). Therefore I ω0 (u n ) I (ω0 +2c)( un (1 ψ n ) ) + O(1).

131 Sketch of the proof: conclusion for ω = ω 0 We reach a contradiction unless J ω0 (u n ψ n ) 0. This implies that ψ n u n ( ξ n ) w k2, where w k2 is the nontrivial minimum of J ω0. With this extra information, we have a better estimate: I ω0 (u n ) 2πξ n J ω0 (u n ψ n ) + I ω0 ( un (1 ψ n ) ) + c u n (1 ψ n ) 2 L 2 (R 2 ) + O(1). Therefore I ω0 (u n ) I (ω0 +2c)( un (1 ψ n ) ) + O(1). We already know that I (ω0 +2c) is bounded from below, and hence inf I ω0 >.

132 On the solutions of (P)

133 On the solutions of (P) Theorem (On the boundedness of I ω ) Let p (1, 3). We have: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive. Theorem (A.P. & D. Ruiz) For almost every ω (0, ω 0 ], (P) admits a positive solution. Moreover, there exist ω > ω > ω 0 such that: if ω > ω, then (P) has no solutions different from zero; if ω (ω 0, ω), then (P) admits at least two positive solutions: one of them is a global minimizer for I ω and the other is a mountain-pass solution.

134 On the solutions of (P) Theorem (On the boundedness of I ω ) Let p (1, 3). We have: if ω (0, ω 0 ), then I ω is unbounded from below; if ω = ω 0, then I ω0 is bounded from below, not coercive and inf I ω0 < 0; if ω > ω 0, then I ω is bounded from below and coercive. Theorem (A.P. & D. Ruiz) For almost every ω (0, ω 0 ], (P) admits a positive solution. Moreover, there exist ω > ω > ω 0 such that: if ω > ω, then (P) has no solutions different from zero; if ω (ω 0, ω), then (P) admits at least two positive solutions: one of them is a global minimizer for I ω and the other is a mountain-pass solution.

135 Sketch of the proof: case ω (0, ω 0 ]

136 Sketch of the proof: case ω (0, ω 0 ] Performing the rescaling u u ω = ω u( ω ), we get [ 1 ( I ω (u ω ) = ω u 2 + u 2) dx + 1 u 2 ( (x) x 2 2 R 2 8 R 2 x 2 su (s)ds) 2 dx 0 ] ω p 3 2 u p+1 dx. p + 1 R 2

137 Sketch of the proof: case ω (0, ω 0 ] Performing the rescaling u u ω = ω u( ω ), we get [ 1 ( I ω (u ω ) = ω u 2 + u 2) dx + 1 u 2 ( (x) x 2 2 R 2 8 R 2 x 2 su (s)ds) 2 dx 0 ] ω p 3 2 u p+1 dx. p + 1 R 2 The geometrical assumptions of the Mountain Pass Theorem are satisfied and we can apply the monotonicity trick [Struwe, Jeanjean], finding a solution for almost every ω (0, ω 0 ].

138 Sketch of the proof: case ω > ω 0 Recall that, for any u H 1 r (R 2 ), u(x) 4 dx R 2 ( ) 1 ( 2 u(x) 2 2 u 2 ( x 2 12 dx R 2 R 2 x 2 su (s)ds) dx) 2. 0

139 Sketch of the proof: case ω > ω 0 Recall that, for any u H 1 r (R 2 ), u(x) 4 dx R 2 ( ) 1 ( 2 u(x) 2 2 u 2 ( x 2 12 dx R 2 R 2 x 2 su (s)ds) dx) 2. 0 Let u be a solution of (P). We multiply (P) by u and integrate. By this inequality, we get 0 1 u 2 dx + (ωu ) 4 u4 u p+1 dx. R 2 R 2

140 Sketch of the proof: case ω > ω 0 Recall that, for any u H 1 r (R 2 ), u(x) 4 dx R 2 ( ) 1 ( 2 u(x) 2 2 u 2 ( x 2 12 dx R 2 R 2 x 2 su (s)ds) dx) 2. 0 Let u be a solution of (P). We multiply (P) by u and integrate. By this inequality, we get 0 1 u 2 dx + (ωu ) 4 u4 u p+1 dx. R 2 R 2 Since there exists ω > 0 such that, for ω > ω, the function t ωt t4 t p+1 is non-negative, then u = 0.

141 Sketch of the proof: case ω > ω0

142 Sketch of the proof: case ω > ω 0 Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω).

143 Sketch of the proof: case ω > ω 0 Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω). Being I ω coercive and weakly lower semicontinuous, the infimum is attained (at negative level).

144 Sketch of the proof: case ω > ω 0 Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω). Being I ω coercive and weakly lower semicontinuous, the infimum is attained (at negative level). If ω (ω 0, ω), the functional satisfies the geometrical assumptions of the Mountain Pass Theorem.

145 Sketch of the proof: case ω > ω 0 Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω). Being I ω coercive and weakly lower semicontinuous, the infimum is attained (at negative level). If ω (ω 0, ω), the functional satisfies the geometrical assumptions of the Mountain Pass Theorem. Since I ω is coercive, (PS) sequences are bounded.

146 Sketch of the proof: case ω > ω 0 Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω). Being I ω coercive and weakly lower semicontinuous, the infimum is attained (at negative level). If ω (ω 0, ω), the functional satisfies the geometrical assumptions of the Mountain Pass Theorem. Since I ω is coercive, (PS) sequences are bounded. We find a second solution (a mountain-pass solution) which is at a positive energy level.

On Chern-Simons-Schrödinger equations including a vortex point

On Chern-Simons-Schrödinger equations including a vortex point On Chern-Simons-Schrödinger equations including a vortex point Alessio Pomponio Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Workshop in Nonlinear PDEs Brussels, September 7

More information

Some results on the nonlinear Klein-Gordon-Maxwell equations

Some results on the nonlinear Klein-Gordon-Maxwell equations Some results on the nonlinear Klein-Gordon-Maxwell equations Alessio Pomponio Dipartimento di Matematica, Politecnico di Bari, Italy Granada, Spain, 2011 A solitary wave is a solution of a field equation

More information

The Chern-Simons-Schrödinger equation

The Chern-Simons-Schrödinger equation The Chern-Simons-Schrödinger equation Low regularity local wellposedness Baoping Liu, Paul Smith, Daniel Tataru University of California, Berkeley July 16, 2012 Paul Smith (UC Berkeley) Chern-Simons-Schrödinger

More information

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS

More information

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume, 998, 83 93 AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS Vieri Benci Donato Fortunato Dedicated to

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

General Relativity and Cosmology Mock exam

General Relativity and Cosmology Mock exam Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers

More information

A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION

A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION JORGE GARCÍA-MELIÁN, JULIO D. ROSSI AND JOSÉ C. SABINA DE LIS Abstract. In this paper we study existence and multiplicity of

More information

Lorentzian elasticity arxiv:

Lorentzian elasticity arxiv: Lorentzian elasticity arxiv:1805.01303 Matteo Capoferri and Dmitri Vassiliev University College London 14 July 2018 Abstract formulation of elasticity theory Consider a manifold M equipped with non-degenerate

More information

CLASSIFICATION OF NON-ABELIAN CHERN-SIMONS VORTICES

CLASSIFICATION OF NON-ABELIAN CHERN-SIMONS VORTICES CLASSIFICATION OF NON-ABELIAN CHERN-SIMONS VORTICES arxiv:hep-th/9310182v1 27 Oct 1993 Gerald V. Dunne Department of Physics University of Connecticut 2152 Hillside Road Storrs, CT 06269 USA dunne@hep.phys.uconn.edu

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3 Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................

More information

Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds

Bielefeld Course on Nonlinear Waves - June 29, Department of Mathematics University of North Carolina, Chapel Hill. Solitons on Manifolds Joint work (on various projects) with Pierre Albin (UIUC), Hans Christianson (UNC), Jason Metcalfe (UNC), Michael Taylor (UNC), Laurent Thomann (Nantes) Department of Mathematics University of North Carolina,

More information

On Schrödinger equations with inverse-square singular potentials

On Schrödinger equations with inverse-square singular potentials On Schrödinger equations with inverse-square singular potentials Veronica Felli Dipartimento di Statistica University of Milano Bicocca veronica.felli@unimib.it joint work with Elsa M. Marchini and Susanna

More information

Blow-up solutions for critical Trudinger-Moser equations in R 2

Blow-up solutions for critical Trudinger-Moser equations in R 2 Blow-up solutions for critical Trudinger-Moser equations in R 2 Bernhard Ruf Università degli Studi di Milano The classical Sobolev embeddings We have the following well-known Sobolev inequalities: let

More information

Electrically and Magnetically Charged Solitons in Gauge Field Th

Electrically and Magnetically Charged Solitons in Gauge Field Th Electrically and Magnetically Charged Solitons in Gauge Field Theory Polytechnic Institute of New York University Talk at the conference Differential and Topological Problems in Modern Theoretical Physics,

More information

Boot camp - Problem set

Boot camp - Problem set Boot camp - Problem set Luis Silvestre September 29, 2017 In the summer of 2017, I led an intensive study group with four undergraduate students at the University of Chicago (Matthew Correia, David Lind,

More information

Nonlinear instability of half-solitons on star graphs

Nonlinear instability of half-solitons on star graphs Nonlinear instability of half-solitons on star graphs Adilbek Kairzhan and Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Workshop Nonlinear Partial Differential Equations on

More information

Uniqueness of ground state solutions of non-local equations in R N

Uniqueness of ground state solutions of non-local equations in R N Uniqueness of ground state solutions of non-local equations in R N Rupert L. Frank Department of Mathematics Princeton University Joint work with Enno Lenzmann and Luis Silvestre Uniqueness and non-degeneracy

More information

Some lecture notes for Math 6050E: PDEs, Fall 2016

Some lecture notes for Math 6050E: PDEs, Fall 2016 Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.

More information

Covariant electrodynamics

Covariant electrodynamics Lecture 9 Covariant electrodynamics WS2010/11: Introduction to Nuclear and Particle Physics 1 Consider Lorentz transformations pseudo-orthogonal transformations in 4-dimentional vector space (Minkowski

More information

Existence of Positive Solutions to a Nonlinear Biharmonic Equation

Existence of Positive Solutions to a Nonlinear Biharmonic Equation International Mathematical Forum, 3, 2008, no. 40, 1959-1964 Existence of Positive Solutions to a Nonlinear Biharmonic Equation S. H. Al Hashimi Department of Chemical Engineering The Petroleum Institute,

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS

REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS fredi tröltzsch 1 Abstract. A class of quadratic optimization problems in Hilbert spaces is considered,

More information

SPECIAL RELATIVITY AND ELECTROMAGNETISM

SPECIAL RELATIVITY AND ELECTROMAGNETISM SPECIAL RELATIVITY AND ELECTROMAGNETISM MATH 460, SECTION 500 The following problems (composed by Professor P.B. Yasskin) will lead you through the construction of the theory of electromagnetism in special

More information

2 A Model, Harmonic Map, Problem

2 A Model, Harmonic Map, Problem ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or

More information

Hardy Rellich inequalities with boundary remainder terms and applications

Hardy Rellich inequalities with boundary remainder terms and applications manuscripta mathematica manuscript No. (will be inserted by the editor) Elvise Berchio Daniele Cassani Filippo Gazzola Hardy Rellich inequalities with boundary remainder terms and applications Received:

More information

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

More information

An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

More information

Topological Solitons from Geometry

Topological Solitons from Geometry Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv:1111.2934.

More information

arxiv: v3 [math.ap] 1 Oct 2018

arxiv: v3 [math.ap] 1 Oct 2018 ON A CLASS OF NONLINEAR SCHRÖDINGER-POISSON SYSTEMS INVOLVING A NONRADIAL CHARGE DENSITY CARLO MERCURI AND TERESA MEGAN TYLER arxiv:1805.00964v3 [math.ap] 1 Oct 018 Abstract. In the spirit of the classical

More information

Bulletin of the. Iranian Mathematical Society

Bulletin of the. Iranian Mathematical Society ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 42 (2016), No. 1, pp. 129 141. Title: On nonlocal elliptic system of p-kirchhoff-type in Author(s): L.

More information

On the relation between scaling properties of functionals and existence of constrained minimizers

On the relation between scaling properties of functionals and existence of constrained minimizers On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini

More information

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.

Übungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor. Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση

More information

Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere

Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere arxiv:math-ph/99126v1 17 Oct 1999 Piotr Bizoń Institute of Physics, Jagellonian University, Kraków, Poland March 26, 28 Abstract

More information

v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0.

v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0. Sep. 26 The Perron Method In this lecture we show that one can show existence of solutions using maximum principle alone.. The Perron method. Recall in the last lecture we have shown the existence of solutions

More information

Liquid crystal flows in two dimensions

Liquid crystal flows in two dimensions Liquid crystal flows in two dimensions Fanghua Lin Junyu Lin Changyou Wang Abstract The paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of

More information

Solutions with prescribed mass for nonlinear Schrödinger equations

Solutions with prescribed mass for nonlinear Schrödinger equations Solutions with prescribed mass for nonlinear Schrödinger equations Dario Pierotti Dipartimento di Matematica, Politecnico di Milano (ITALY) Varese - September 17, 2015 Work in progress with Gianmaria Verzini

More information

The Schrödinger Poisson equation under the effect of a nonlinear local term

The Schrödinger Poisson equation under the effect of a nonlinear local term Journal of Functional Analysis 237 (2006) 655 674 www.elsevier.com/locate/jfa The Schrödinger Poisson equation under the effect of a nonlinear local term David Ruiz Departamento de Análisis Matemático,

More information

Quasi-neutral limit for Euler-Poisson system in the presence of plasma sheaths

Quasi-neutral limit for Euler-Poisson system in the presence of plasma sheaths in the presence of plasma sheaths Department of Mathematical Sciences Ulsan National Institute of Science and Technology (UNIST) joint work with Masahiro Suzuki (Nagoya) and Chang-Yeol Jung (Ulsan) The

More information

Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations

Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations Bound-state solutions and well-posedness of the dispersion-managed nonlinear Schrödinger and related equations J. Albert and E. Kahlil University of Oklahoma, Langston University 10th IMACS Conference,

More information

DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS

DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN Abstract. When the coefficients of the cubic terms match the coefficients in the boundary

More information

Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory

Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou National Center of Theoretical Science December 19, 2015

More information

Non-radial solutions to a bi-harmonic equation with negative exponent

Non-radial solutions to a bi-harmonic equation with negative exponent Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei

More information

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00

MSci EXAMINATION. Date: XX th May, Time: 14:30-17:00 MSci EXAMINATION PHY-415 (MSci 4242 Relativistic Waves and Quantum Fields Time Allowed: 2 hours 30 minutes Date: XX th May, 2010 Time: 14:30-17:00 Instructions: Answer THREE QUESTIONS only. Each question

More information

Phase-field systems with nonlinear coupling and dynamic boundary conditions

Phase-field systems with nonlinear coupling and dynamic boundary conditions 1 / 46 Phase-field systems with nonlinear coupling and dynamic boundary conditions Cecilia Cavaterra Dipartimento di Matematica F. Enriques Università degli Studi di Milano cecilia.cavaterra@unimi.it VIII

More information

Symmetry breaking for a problem in optimal insulation

Symmetry breaking for a problem in optimal insulation Symmetry breaking for a problem in optimal insulation Giuseppe Buttazzo Dipartimento di Matematica Università di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it Geometric and Analytic Inequalities Banff,

More information

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.

Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological

More information

Neoclassical Theory of Electromagnetic Interactions II

Neoclassical Theory of Electromagnetic Interactions II Neoclassical Theory of Electromagnetic Interactions II One theory for all scales Alexander Figotin and Anatoli Babin University of California at Irvine The work was supported by AFOSR July, 2016 A. Figotin

More information

Booklet of Abstracts Brescia Trento Nonlinear Days Second Edition 25th May 2018

Booklet of Abstracts Brescia Trento Nonlinear Days Second Edition 25th May 2018 Booklet of Abstracts Brescia Trento Nonlinear Days Second Edition 25th May 2018 2 Recent updates on double phase variational integrals Paolo Baroni, Università di Parma Abstract: I will describe some recent

More information

Gravitation: Tensor Calculus

Gravitation: Tensor Calculus An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013

More information

Spinor Formulation of Relativistic Quantum Mechanics

Spinor Formulation of Relativistic Quantum Mechanics Chapter Spinor Formulation of Relativistic Quantum Mechanics. The Lorentz Transformation of the Dirac Bispinor We will provide in the following a new formulation of the Dirac equation in the chiral representation

More information

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using

More information

Special and General Relativity (PHZ 4601/5606) Fall 2018 Classwork and Homework. Every exercise counts 10 points unless stated differently.

Special and General Relativity (PHZ 4601/5606) Fall 2018 Classwork and Homework. Every exercise counts 10 points unless stated differently. 1 Special and General Relativity (PHZ 4601/5606) Fall 2018 Classwork and Homework Every exercise counts 10 points unless stated differently. Set 1: (1) Homework, due ( F ) 8/31/2018 before ( ) class. Consider

More information

TRANSPORT IN POROUS MEDIA

TRANSPORT IN POROUS MEDIA 1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case

More information

The Helically Reduced Wave Equation as a Symmetric Positive System

The Helically Reduced Wave Equation as a Symmetric Positive System Utah State University DigitalCommons@USU All Physics Faculty Publications Physics 2003 The Helically Reduced Wave Equation as a Symmetric Positive System Charles G. Torre Utah State University Follow this

More information

Exercise 1 Classical Bosonic String

Exercise 1 Classical Bosonic String Exercise 1 Classical Bosonic String 1. The Relativistic Particle The action describing a free relativistic point particle of mass m moving in a D- dimensional Minkowski spacetime is described by ) 1 S

More information

APMA 2811Q. Homework #1. Due: 9/25/13. 1 exp ( f (x) 2) dx, I[f] =

APMA 2811Q. Homework #1. Due: 9/25/13. 1 exp ( f (x) 2) dx, I[f] = APMA 8Q Homework # Due: 9/5/3. Ill-posed problems a) Consider I : W,, ) R defined by exp f x) ) dx, where W,, ) = f W,, ) : f) = f) = }. Show that I has no minimizer in A. This problem is not coercive

More information

SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS

SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties

More information

Existence and stability of solitary-wave solutions to nonlocal equations

Existence and stability of solitary-wave solutions to nonlocal equations Existence and stability of solitary-wave solutions to nonlocal equations Mathias Nikolai Arnesen Norwegian University of Science and Technology September 22nd, Trondheim The equations u t + f (u) x (Lu)

More information

221A Miscellaneous Notes Continuity Equation

221A Miscellaneous Notes Continuity Equation 221A Miscellaneous Notes Continuity Equation 1 The Continuity Equation As I received questions about the midterm problems, I realized that some of you have a conceptual gap about the continuity equation.

More information

Geometric inequalities for black holes

Geometric inequalities for black holes Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with

More information

An introduction to Birkhoff normal form

An introduction to Birkhoff normal form An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

More information

Late-time tails of self-gravitating waves

Late-time tails of self-gravitating waves Late-time tails of self-gravitating waves (non-rigorous quantitative analysis) Piotr Bizoń Jagiellonian University, Kraków Based on joint work with Tadek Chmaj and Andrzej Rostworowski Outline: Motivation

More information

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute

More information

OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1.

OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1. OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV Abstract. We consider the Schrödinger operator + V (x) on H 0 (), where is a given domain of R d. Our

More information

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC

More information

Lecture No 1 Introduction to Diffusion equations The heat equat

Lecture No 1 Introduction to Diffusion equations The heat equat Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and

More information

arxiv: v2 [math.ap] 28 Nov 2016

arxiv: v2 [math.ap] 28 Nov 2016 ONE-DIMENSIONAL SAIONARY MEAN-FIELD GAMES WIH LOCAL COUPLING DIOGO A. GOMES, LEVON NURBEKYAN, AND MARIANA PRAZERES arxiv:1611.8161v [math.ap] 8 Nov 16 Abstract. A standard assumption in mean-field game

More information

Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations

Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455

More information

Shape optimization problems for variational functionals under geometric constraints

Shape optimization problems for variational functionals under geometric constraints Shape optimization problems for variational functionals under geometric constraints Ilaria Fragalà 2 nd Italian-Japanese Workshop Cortona, June 20-24, 2011 The variational functionals The first Dirichlet

More information

Nonlinear problems with lack of compactness in Critical Point Theory

Nonlinear problems with lack of compactness in Critical Point Theory Nonlinear problems with lack of compactness in Critical Point Theory Carlo Mercuri CASA Day Eindhoven, 11th April 2012 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here

More information

SEQUENCES OF SMALL HOMOCLINIC SOLUTIONS FOR DIFFERENCE EQUATIONS ON INTEGERS

SEQUENCES OF SMALL HOMOCLINIC SOLUTIONS FOR DIFFERENCE EQUATIONS ON INTEGERS Electronic Journal of Differential Equations, Vol. 2017 (2017, No. 228, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SEQUENCES OF SMALL HOMOCLINIC SOLUTIONS

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

On the Schrödinger Equation in R N under the Effect of a General Nonlinear Term

On the Schrödinger Equation in R N under the Effect of a General Nonlinear Term On the Schrödinger Equation in under the Effect of a General Nonlinear Term A. AZZOLLINI & A. POMPONIO ABSTRACT. In this paper we prove the existence of a positive solution to the equation u + V(x)u =

More information

2.3 Calculus of variations

2.3 Calculus of variations 2.3 Calculus of variations 2.3.1 Euler-Lagrange equation The action functional S[x(t)] = L(x, ẋ, t) dt (2.3.1) which maps a curve x(t) to a number, can be expanded in a Taylor series { S[x(t) + δx(t)]

More information

Nonlinear scalar field equations with general nonlinearity

Nonlinear scalar field equations with general nonlinearity Louis Jeanjean Laboratoire de Mathématiques (CNRS UMR 6623) Université de Bourgogne Franche-Comté Besançon, 253, France e-mail: louis.jeanjean@univ-fcomte.fr Sheng-Sen Lu Center for Applied Mathematics,

More information

Vortex solutions of the Liouville equation

Vortex solutions of the Liouville equation Vortex solutions of the Liouville equation P. A. HORVÁTHY and J.-C. YÉRA Laboratoire de Mathématiques et de Physique Théorique arxiv:hep-th/9805161v1 25 May 1998 Université de Tours Parc de Grandmont,

More information

Partial Differential Equations

Partial Differential Equations Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,

More information

A minimization problem for the Nonlinear

A minimization problem for the Nonlinear São Paulo Journal of Mathematical Sciences 5, 2 (211), 149 173 A minimization problem for the Nonlinear Schrödinger-Poisson type Equation Gaetano Siciliano Dipartimento di Matematica, Università degli

More information

Hylomorphic solitons and their dynamics

Hylomorphic solitons and their dynamics Hylomorphic solitons and their dynamics Vieri Benci Dipartimento di Matematica Applicata U. Dini Università di Pisa 18th May 2009 Vieri Benci (DMA-Pisa) Hylomorphic solitons 18th May 2009 1 / 50 Types

More information

Nonlinear Wave Equations and Solitary Wave Solutions in Mathematical Physics

Nonlinear Wave Equations and Solitary Wave Solutions in Mathematical Physics Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2012 Nonlinear Wave Equations and Solitary Wave Solutions in Mathematical Physics Trevor Caldwell Harvey Mudd College

More information

RANDOM FIELDS AND GEOMETRY. Robert Adler and Jonathan Taylor

RANDOM FIELDS AND GEOMETRY. Robert Adler and Jonathan Taylor RANDOM FIELDS AND GEOMETRY from the book of the same name by Robert Adler and Jonathan Taylor IE&M, Technion, Israel, Statistics, Stanford, US. ie.technion.ac.il/adler.phtml www-stat.stanford.edu/ jtaylor

More information

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy

More information

Physics 236a assignment, Week 2:

Physics 236a assignment, Week 2: Physics 236a assignment, Week 2: (October 8, 2015. Due on October 15, 2015) 1. Equation of motion for a spin in a magnetic field. [10 points] We will obtain the relativistic generalization of the nonrelativistic

More information

Minimal Surfaces: Nonparametric Theory. Andrejs Treibergs. January, 2016

Minimal Surfaces: Nonparametric Theory. Andrejs Treibergs. January, 2016 USAC Colloquium Minimal Surfaces: Nonparametric Theory Andrejs Treibergs University of Utah January, 2016 2. USAC Lecture: Minimal Surfaces The URL for these Beamer Slides: Minimal Surfaces: Nonparametric

More information

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL

More information

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

More information

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC

More information

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

Global existence and compact attractors for the discrete nonlinear Schrödinger equation J. Differential Equations 7 (005) 88 3 www.elsevier.com/locate/jde Global existence and compact attractors for the discrete nonlinear Schrödinger equation Nikos I. Karachalios a,, Athanasios N. Yannacopoulos

More information

LOCALIZATION OF FIELDS ON A BRANE IN SIX DIMENSIONS.

LOCALIZATION OF FIELDS ON A BRANE IN SIX DIMENSIONS. LOCALIZATION OF FIELDS ON A BRANE IN SIX DIMENSIONS Merab Gogberashvili a and Paul Midodashvili b a Andronikashvili Institute of Physics, 6 Tamarashvili Str., Tbilisi 3877, Georgia E-mail: gogber@hotmail.com

More information

ODE solutions for the fractional Laplacian equations arising in co

ODE solutions for the fractional Laplacian equations arising in co ODE solutions for the fractional Laplacian equations arising in conformal geometry Granada, November 7th, 2014 Background Classical Yamabe problem (M n, g) = Riemannian manifold; n 3, g = u 4 n 2 g conformal

More information

Non-existence of time-periodic vacuum spacetimes

Non-existence of time-periodic vacuum spacetimes Non-existence of time-periodic vacuum spacetimes Volker Schlue (joint work with Spyros Alexakis and Arick Shao) Université Pierre et Marie Curie (Paris 6) Dynamics of self-gravitating matter workshop,

More information

Twistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/

Twistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/ Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/0606272 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry Amplitudes for YM, Gravity have elegant

More information

On some weighted fractional porous media equations

On some weighted fractional porous media equations On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME

More information

The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant

The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant Jared Speck & Igor Rodnianski jspeck@math.princeton.edu University of Cambridge & Princeton University October

More information

Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N

Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N Threshold solutions and sharp transitions for nonautonomous parabolic equations on R N P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract This paper is devoted to

More information

Physics 506 Winter 2006 Homework Assignment #8 Solutions

Physics 506 Winter 2006 Homework Assignment #8 Solutions Physics 506 Winter 2006 Homework Assignment #8 Solutions Textbook problems: Ch. 11: 11.13, 11.16, 11.18, 11.27 11.13 An infinitely long straight wire of negligible cross-sectional area is at rest and has

More information

Part IV: Numerical schemes for the phase-filed model

Part IV: Numerical schemes for the phase-filed model Part IV: Numerical schemes for the phase-filed model Jie Shen Department of Mathematics Purdue University IMS, Singapore July 29-3, 29 The complete set of governing equations Find u, p, (φ, ξ) such that

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information