On ChernSimonsSchrödinger equations including a vortex point


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1 On ChernSimonsSchrödinger equations including a vortex point Alessio Pomponio Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Workshop in Nonlinear PDEs Brussels, September 7 11, 2015.
2 We are concerned with the ChernSimonsSchrödinger equations: 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), (1) 0 A 2 2 A 0 = Im( φd 1 φ), 1 A 2 2 A 1 = 1 2 φ 2. Here t R, x = (x 1, x 2 ) R 2 and the unknowns are (φ, A 0, A 1, A 2 ), where 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 This model was first proposed and studied by Jackiw & Pi, in several papers in the beginning of 90s in order to describe nonrelativistic matter interacting with ChernSimons gauge fields in the plane.
4 This model was first proposed and studied by Jackiw & Pi, in several papers in the beginning of 90s in order to describe nonrelativistic matter interacting with ChernSimons gauge fields in the plane. The initial value problem, as well as global existence and blowup, has been addressed in [Bergé, de Bouard & Saut, 1995; Huh, ] for the case p = 3.
5 The existence of stationary states for (1) and general p > 1 has been studied in [Byeon, Huh & Seok, JFA 2012] for the regular case.
6 The existence of stationary states for (1) and general p > 1 has been studied in [Byeon, Huh & Seok, JFA 2012] for the regular case. 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.
7 More recently, the existence of stationary states for (1) with a vortex point of order N, for an arbitrary N N has been considered in [Byeon, Huh & Seok, 2014].
8 More recently, the existence of stationary states for (1) with a vortex point of order N, for an arbitrary N N has been considered in [Byeon, Huh & Seok, 2014]. They are no more interested in 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 ).
9 More recently, the existence of stationary states for (1) with a vortex point of order N, for an arbitrary N N has been considered in [Byeon, Huh & Seok, 2014]. They seek the standing wave solutions of the form φ(t, x) = u( x )e i(nθ+ωt), 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 tan θ = x 2 /x 1.
10 With this ansatz, first, they solve A 0, A 1, A 2 in terms of u.
11 With this ansatz, first, they solve A 0, A 1, A 2 in terms of u. They find that h u (s) N A 0 (x) = ξ + u 2 (s) ds, x s where ξ appears as an integration constant and h u (r) = r 0 s 2 u2 (s) ds.
12 With this ansatz, first, they solve A 0, A 1, A 2 in terms of u. They find that h u (s) N A 0 (x) = ξ + u 2 (s) ds, x s where ξ appears as an integration constant and Moreover h u (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.
13 Therefore one need only to solve, in R 2, the equation: u + (ω + ξ + (h u( x ) N) 2 + x 2 + x ) h u (s) N u 2 (s) ds u s = u p 1 u.
14 Therefore one need only to solve, in R 2, the equation: u + (ω + ξ + (h u( x ) N) 2 + x 2 + x ) h u (s) N u 2 (s) ds u s = u p 1 u. Observe that the constant ω + ξ is a gauge invariant of the stationary solutions of the problem.
15 The equation So 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].
16 The equation So 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 + (h u( x ) N) 2 + x 2 + x ) h u (s) N u 2 (s) ds u s = u p 1 u. (CSS) where h u (r) = r 0 s 2 u2 (s) ds.
17 The case N = 0 In [Byeon, Huh, Seok, JFA 2012] it is shown that (CSS) is indeed the EulerLagrange 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
18 The case N = 0 In [Byeon, Huh, Seok, JFA 2012] it is shown that (CSS) is indeed the EulerLagrange 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
19 The case N = 0 In [Byeon, Huh, Seok, JFA 2012] it is shown that (CSS) is indeed the EulerLagrange 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 The nonlocal term is well defined in H 1 r (R 2 ).
20 The case N = 0 When N = 0, formally (CSS) is the EulerLagrange equation of the energy functional 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 2N dx 1 u p+1 dx. 0 p + 1 R 2
21 The case N = 0 When N = 0, formally (CSS) is the EulerLagrange equation of the energy functional 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 2N dx 1 u p+1 dx. 0 p + 1 R 2
22 The case N = 0 The term u 2 (x) R 2 x 2 ( x 2 su 2 (s) ds 2N) dx 0 is not well defined in H 1 r (R 2 ), indeed, in particular, it contains 4N 2 u 2 (x) R 2 x 2 dx.
23 The case N = 0: the space H The functional I ω is well defined in H is defined as endowed by the norm H = {u Hr 1 (R 2 u ) 2 (x) : R 2 x 2 dx < + }, u 2 H = R 2 u(x) 2 + ( ) x 2 u 2 (x) dx.
24 The case N = 0: the space H The functional I ω is well defined in H is defined as endowed by the norm H = {u Hr 1 (R 2 u ) 2 (x) : R 2 x 2 dx < + }, u 2 H = R 2 u(x) 2 + ( ) x 2 u 2 (x) dx. In [Byeon, Huh & Seok, 2014], it is shown that H {u C(R 2 ) : u(0) = 0} L (R 2 ).
25 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term.
26 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below.
27 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountainpass geometry.
28 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountainpass 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.
29 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountainpass 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, if N = 0, in [Byeon, Huh & Seok, JFA2012].
30 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountainpass 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, if N = 0, in [Byeon, Huh & Seok, JFA2012]. If, instead, N = 0, the approach is based on the monotonicity trick, in [Byeon, Huh & Seok, 2014].
31 ByeonHuhSeok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term. I ω is unbounded from below. I ω exhibits a mountainpass 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, if N = 0, in [Byeon, Huh & Seok, JFA2012]. If, instead, N = 0, the approach is based on the monotonicity trick, in [Byeon, Huh & Seok, 2014]. Infinitely many (possibly signchanging) solutions have been found in [Huh, JMP 2012] for p > 5 and N = 0: this case is more easy since (PS)condition holds.
32 ByeonHuhSeok results: case p = 3 This is a special case:
33 ByeonHuhSeok results: case p = 3 This is a special case: for any N N {0}, static solutions can be found by passing to a selfdual equation, which leads to a Liouville equation that can be solved explicitly.
34 ByeonHuhSeok results: case p = 3 This is a special case: for any N N {0}, static solutions can be found by passing to a selfdual 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 have the following form: (φ, A 0, A 1, A 2 ) = ( 8l(N + 1) lx N e i(nθ+ωt) 1 + lx 2(N+1), where l > 0 is an arbitrary real constant. ( ) 2l(N + 1) lx N lx 2(N+1) ω, ), 2l 2 (N + 1)x 2 lx 2N 1 + lx 2(N+1), 2l2 (N + 1)x 1 lx 2N 1 + lx 2(N+1)
35 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity.
36 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, solutions are found as minimizers on a L 2 sphere.
37 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, solutions are found as minimizers on a L 2 sphere. The value ω comes out as a Lagrange multiplier, and it is not controlled.
38 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, 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).
39 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, 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 (CSS), a solution u is found only for a particular value of ω.
40 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, 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 (CSS), a solution u is found only for a particular value of ω. The global behavior of the energy functional I ω is not studied.
41 ByeonHuhSeok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity. For N = 0, 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 (CSS), a solution u is found only for a particular value of ω. The global behavior of the energy functional I ω is not studied. The case N = 0 is not treated.
42 On the boundedness from below of Iω
43 On the boundedness from below of I ω Theorem (Y. Jiang, A.P. & D. Ruiz) Let N N {0} and p (1, 3).
44 On the boundedness from below of I ω Theorem (Y. Jiang, A.P. & D. Ruiz) Let N N {0} and p (1, 3). There exists ω 0 such that: if ω (0, ω 0 ), then I ω is unbounded from below;
45 On the boundedness from below of I ω Theorem (Y. Jiang, A.P. & D. Ruiz) Let N N {0} and 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;
46 On the boundedness from below of I ω Theorem (Y. Jiang, A.P. & D. Ruiz) Let N N {0} and 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.
47 On the boundedness from below of I ω Theorem (Y. Jiang, A.P. & D. Ruiz) Let N N {0} and 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.
48 Rough sketch of the proof I ω is coercive when the problem is posed on a bounded domain.
49 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.
50 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 } n, the problem is the possible loss of mass at infinity as n +. We need to study the behavior of those masses.
51 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 } 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 } n behaves as a soliton, if u n is interpreted as a function of a single real variable.
52 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 } 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 } 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.
53 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 } 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 } 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.
54 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, and define u ρ (r) = u(r ρ). Let us now estimate I ω (u ρ ) as ρ +.
55 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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 2N) dr u ρ p+1 r dr.
56 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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 2N) dr ρ u p+1 (r + ρ) dr.
57 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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 2N) dr u p+1 (r + ρ) dr.
58 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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 2N) 2 dr
59 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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 2N) 2 dr
60 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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.
61 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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.
62 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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.
63 The limit functional Let u be a fixed even function which decays exponentially to zero at infinity, 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.
64 For any N N {0}, 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
65 For any N N {0}, 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 +.
66 For any N N {0}, 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 =.
67 For any N N {0}, 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 =. Actually, we can show that inf J ω < 0 inf I ω =.
68 The limit functional Proposition Let p (1, 3) and ω > 0. Then: a) J ω is coercive and attains its infimum;
69 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 ;
70 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 ).
71 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 ). The critical points of J ω, and their energy, can be found explicitly, so we can find ω 0.
72 On the solutions of (CSS), for any N N {0}
73 On the solutions of (CSS), for any N N {0} 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 (Y. Jiang, A.P. & D. Ruiz) For almost every ω (0, ω 0 ], (CSS) admits a positive solution. Moreover, there exist ω > ω > ω 0 such that: if ω > ω, then (CSS) has no solutions different from zero; if ω (ω 0, ω), then (CSS) admits at least two positive solutions: one of them is a global minimizer for I ω and the other is a mountainpass solution.
74 On the solutions of (CSS), for any N N {0} 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 (Y. Jiang, A.P. & D. Ruiz) For almost every ω (0, ω 0 ], (CSS) admits a positive solution. Moreover, there exist ω > ω > ω 0 such that: if ω > ω, then (CSS) has no solutions different from zero; if ω (ω 0, ω), then (CSS) admits at least two positive solutions: one of them is a global minimizer for I ω and the other is a mountainpass solution.
75 Sketch of the proof: case ω (0, ω 0 ]
76 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 2 R u 2 ( (x) x ) 2 ] 8 R 2 x 2 su 2 (s)ds 2N dx ω p 3 2 u p+1 dx. 0 p + 1 R 2
77 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 2 R u 2 ( (x) x ) 2 ] 8 R 2 x 2 su 2 (s)ds 2N dx ω p 3 2 u p+1 dx. 0 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 ].
78 Sketch of the proof: case ω > ω0
79 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero.
80 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero. Since inf I ω0 < 0, there exists ω > ω 0 such that inf I ω < 0 for ω (ω 0, ω).
81 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero. 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).
82 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero. 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.
83 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero. 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.
84 Sketch of the proof: case ω > ω 0 By Pohozaev identity arguments, we infer that there exists ω > ω 0 such that, if ω > ω, then (CSS) has no solutions different from zero. 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 mountainpass solution) which is at a positive energy level.
85 Thank you for your attention!!!
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