On Chern-Simons-Schrödinger equations including a vortex point

Transcription

1 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 11, 2015.

2 We are concerned with the Chern-Simons-Schrödinger 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 Chern-Simons 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 Chern-Simons gauge fields in the plane. 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.

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 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

18 The case N = 0 In [Byeon, Huh, Seok, JFA 2012] it is shown that (CSS) 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

19 The case N = 0 In [Byeon, Huh, Seok, JFA 2012] it is shown that (CSS) 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 The nonlocal term is well defined in H 1 r (R 2 ).

20 The case N = 0 When N = 0, formally (CSS) is the Euler-Lagrange 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 Euler-Lagrange 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 Byeon-Huh-Seok results: case p > 3 In this case the local nonlinearity dominates the nonlocal term.

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

27 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.

28 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.

29 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, if N = 0, in [Byeon, Huh & Seok, JFA2012].

30 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, 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 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, 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 sign-changing) solutions have been found in [Huh, JMP 2012] for p > 5 and N = 0: this case is more easy since (PS)-condition holds.

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

33 Byeon-Huh-Seok results: case p = 3 This is a special case: for any N N {0}, static solutions can be found by passing to a self-dual equation, which leads to a Liouville equation that can be solved explicitly.

34 Byeon-Huh-Seok results: case p = 3 This is a special case: for any N N {0}, 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 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 Byeon-Huh-Seok results: case 1 < p < 3 In this case, the nonlocal term prevails over the local nonlinearity.

36 Byeon-Huh-Seok 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 Byeon-Huh-Seok 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 Byeon-Huh-Seok 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 Byeon-Huh-Seok 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 Byeon-Huh-Seok 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 Byeon-Huh-Seok 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 mountain-pass 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 mountain-pass 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 mountain-pass solution) which is at a positive energy level.

85 Thank you for your attention!!!