Nonlinear problems with lack of compactness in Critical Point Theory

Transcription

1 Nonlinear problems with lack of compactness in Critical Point Theory Carlo Mercuri CASA Day Eindhoven, 11th April 2012

2 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here P : X Y, where X, Y are Banach Spaces.

3 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here P : X Y, where X, Y are Banach Spaces. Variational formulation when φ = P, φ : X R and φ(u + tv) φ(u) < P(u), v >= lim. t 0 t Here Y = X the topological dual and

4 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here P : X Y, where X, Y are Banach Spaces. Variational formulation when φ = P, φ : X R and φ(u + tv) φ(u) < P(u), v >= lim. t 0 t Here Y = X the topological dual and (1) φ (u) = 0

5 Critical points Many linear and nonlinear PDE s have the form P(u) = 0, u X. (1) Here P : X Y, where X, Y are Banach Spaces. Variational formulation when φ = P, φ : X R and φ(u + tv) φ(u) < P(u), v >= lim. t 0 t Here Y = X the topological dual and (1) φ (u) = 0 i.e. solutions of (1) are critical points of φ.

6 How to find critical points If φ bdd from below, search for c := inf X φ

7 How to find critical points If φ bdd from below, search for c := inf X φ Is c critical value? Namely, c = φ(u), for some u?

8 How to find critical points If φ bdd from below, search for c := inf X φ Is c critical value? Namely, c = φ(u), for some u? By Ekeland s variational principle there exists (u n ) φ(u n ) c φ (u n ) 0 (2)

9 How to find critical points If φ bdd from below, search for c := inf X φ Is c critical value? Namely, c = φ(u), for some u? By Ekeland s variational principle there exists (u n ) φ(u n ) c φ (u n ) 0 (2) i.e. Palais-Smale sequence at level c. Denoted by (PS) c.

10 The Palais-Smale condition If for some subsequence u n ū 0, (3) we say (u n ) satisfies the Palais-Smale condition at level c.

11 The Palais-Smale condition If for some subsequence u n ū 0, (3) we say (u n ) satisfies the Palais-Smale condition at level c. Hence, if φ C 1, c = φ(ū) and c critical.

12 The Palais-Smale condition If for some subsequence u n ū 0, (3) we say (u n ) satisfies the Palais-Smale condition at level c. Hence, if φ C 1, c = φ(ū) and c critical. Problem solved!

13 The (PS) c condition as a compactness condition Take f : K R, bounded and continuous, K closed and bounded in R N. Attains minimum value (Weierstrass).

14 The (PS) c condition as a compactness condition Take f : K R, bounded and continuous, K closed and bounded in R N. Attains minimum value (Weierstrass). Pick (x n ): f (x n ) λ := inf x K f (x)

15 The (PS) c condition as a compactness condition Take f : K R, bounded and continuous, K closed and bounded in R N. Attains minimum value (Weierstrass). Pick (x n ): f (x n ) λ := inf x K f (x) K is compact (Heine-Borel): by Bolzano-Weierstrass for a subs. x n x K

16 The (PS) c condition as a compactness condition Take f : K R, bounded and continuous, K closed and bounded in R N. Attains minimum value (Weierstrass). Pick (x n ): f (x n ) λ := inf x K f (x) K is compact (Heine-Borel): by Bolzano-Weierstrass for a subs. x n x K Since f is continuous:. λ := lim n f (x n ) = f (x)

17 The (PS) c condition as a compactness condition Take f : K R, bounded and continuous, K closed and bounded in R N. Attains minimum value (Weierstrass). Pick (x n ): f (x n ) λ := inf x K f (x) K is compact (Heine-Borel): by Bolzano-Weierstrass for a subs. x n x K Since f is continuous:. λ := lim n f (x n ) = f (x) We notice Palais-Smale condition Bolzano-Weierstrass theorem, in infinite dimensional spaces.

18 Saddle points Consider J not bdd from below? saddle-points.

19 Saddle points Consider J not bdd 8.1from Thebelow? mountain saddle-points. pass theorem 117 Assumption 1: suitable geometry J(tu) t u tu

20 Saddle points Consider J not bdd 8.1from Thebelow? mountain saddle-points. pass theorem 117 Assumption 1: suitable geometry J(tu) t u tu 0 local minimum and as t, J(tu) < 0.

21 The Mountain-Pass theorem (Ambrosetti-Rabinowitz (1973)) Assume J(e) < 0.

22 The Mountain-Pass theorem (Ambrosetti-Rabinowitz (1973)) Assume J(e) < 0 (0, J(0)) and (e, J(e)) separated by a ring of mountains:.

23 The Mountain-Pass theorem (Ambrosetti-Rabinowitz (1973)) Assume J(e) < 0 (0, J(0)) and (e, J(e)) separated by a ring of mountains: Define Γ as the set of paths joining 0 to e, then define. c := inf γ Γ max t [0,1] J(γ(t))

24 Is c a critical point? By Ekeland s variational principle there exists (u n ) J(u n ) c J (u n ) 0

25 Is c a critical point? By Ekeland s variational principle there exists (u n ) J(u n ) c J (u n ) 0 Mountain-Pass theorem: If the (PS) c condition holds (Assumption 2), then c is a critical value.

26 Is c a critical point? By Ekeland s variational principle there exists (u n ) J(u n ) c J (u n ) 0 Mountain-Pass theorem: If the (PS) c condition holds (Assumption 2), then c is a critical value. The Mountain-Pass theorem yields nonzero solutions! Indeed: 0 < inf J(u) c max J(te) u =r t [0,1]

27 The 1,000,000 Dollars (or more) intuition! Define the sublevels M a := {u X : J(u) a}.

28 The 1,000,000 Dollars (or more) intuition! Define the sublevels M a := {u X : J(u) a}. If in [a, b] no critical values, then M a can be deformed into M b.

29 The 1,000,000 Dollars (or more) intuition! Define the sublevels M a := {u X : J(u) a}. If in [a, b] no critical values, then M a can be deformed into M b. If X, (, ) is a Hilbert Space and J C 2 (X ; R), we can deform M a into M b via the gradient flow given by d dtσ(t, u) = J(σ(t, u)) σ(0, u) = u. (4)

30 The 1,000,000 Dollars (or more) intuition! Define the sublevels M a := {u X : J(u) a}. If in [a, b] no critical values, then M a can be deformed into M b. If X, (, ) is a Hilbert Space and J C 2 (X ; R), we can deform M a into M b via the gradient flow given by d dtσ(t, u) = J(σ(t, u)) σ(0, u) = u. σ( t, M b ) M a, and σ(t, ) homeomorphism. (4)

31 z Critical values in [a, b] prevent M a to be deformed into M b. c 4 c 3 p 4 p 4 b c 2 a p 2 c 1 p 1 y x

32 z Critical values in [a, b] prevent M a to be deformed into M b. Example: X R 3 be the torus and J(x, y, z) = z. c 4 c 3 p 4 p 4 b c 2 a p 2 c 1 p 1 y x

33 z Critical values in [a, b] prevent M a to be deformed into M b. Example: X R 3 be the torus and J(x, y, z) = z. Critical points of J: p i where J = (0, 0, 1) X. c 4 c 3 b p 4 p 4 c 2 a p 2 c 1 p 1 y x

34 z Critical values in [a, b] prevent M a to be deformed into M b. Example: X R 3 be the torus and J(x, y, z) = z. Critical points of J: p i where J = (0, 0, 1) X. M b are to a cylinder S 1 [0, 1] if b (c 2, c 3 ). If a (c 1, c 2 ) then M a B 1 R 2. Hence M b cannot be deformed into M a. c 4 p 4 c 3 p 4 b c 2 p 2 a c 1 p 1 y x

35 The method Following [M. Willem, 1983] in order to find critical points:

36 The method Following [M. Willem, 1983] in order to find critical points: Construction of "a"(ps) c sequence (via deformation argument)

37 The method Following [M. Willem, 1983] in order to find critical points: Construction of "a"(ps) c sequence (via deformation argument) A posteriori, compactness condition,

38 The method Following [M. Willem, 1983] in order to find critical points: Construction of "a"(ps) c sequence (via deformation argument) A posteriori, compactness condition, usually c := inf γ Γ max t [0,1] J(γ(t)).

39 The method Following [M. Willem, 1983] in order to find critical points: Construction of "a"(ps) c sequence (via deformation argument) A posteriori, compactness condition, usually c := inf γ Γ max t [0,1] J(γ(t)). Before [M. Willem, 1983] authors used to prove (PS) c condition for every (PS) c seq. s (i.e. a priori).

40 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5)

41 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if:

42 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if: In φ (u) = 0 there are critical nonlinearities

43 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if: In φ (u) = 0 there are critical nonlinearities Noncompact symmetry group for φ (u) = 0 (translation, dilation invariance)

44 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if: In φ (u) = 0 there are critical nonlinearities Noncompact symmetry group for φ (u) = 0 (translation, dilation invariance) Non-compactness of the domain for φ (u) = 0

45 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if: In φ (u) = 0 there are critical nonlinearities Noncompact symmetry group for φ (u) = 0 (translation, dilation invariance) Non-compactness of the domain for φ (u) = 0 Difficulties in proving (u n ) bounded

46 Verifying the (PS) c condition Pick a (PS) c sequence (u n ) for φ : X R, i.e. : φ(u n ) c φ (u n ) 0 (5) Possible failure of (PS) c condition may occur if: In φ (u) = 0 there are critical nonlinearities Noncompact symmetry group for φ (u) = 0 (translation, dilation invariance) Non-compactness of the domain for φ (u) = 0 Difficulties in proving (u n ) bounded They occur in many interesting problems!

47 Critical nonlinearities, dilation invariance In [C. M. and M. Willem, DCDS-A 2010)] we studied

48 Critical nonlinearities, dilation invariance In [C. M. and M. Willem, DCDS-A 2010)] we studied p u + a(x)u p 1 = µu p 1 in Ω, u 0 in Ω, u = 0 on Ω where Ω smooth and bounded, a( ) given function, µ > 0. Notation p u := div( u p 2 u), 1 < p <, and p := Np/(N p), 1 < p < N.

49 Previous results In [H. Brezis and L. Nirenberg, 1983] and [M. Struwe, 1984] the Laplacian case p = 2

50 Previous results In [H. Brezis and L. Nirenberg, 1983] and [M. Struwe, 1984] the Laplacian case p = 2 Similar nonlinearities in Conformal Geometry (Yamabe problem) and in Quantum Physics (Yang Mills equations)

51 Previous results In [H. Brezis and L. Nirenberg, 1983] and [M. Struwe, 1984] the Laplacian case p = 2 Similar nonlinearities in Conformal Geometry (Yamabe problem) and in Quantum Physics (Yang Mills equations) After > 25 years Theorem (C.M.- M. Willem, 2010) Let 1 < p < N and let u : R N + 0 be nonnegative solution of the equation zero on boundary. Then u 0. p u = µu p 1 in R N +,

52 Previous results In [H. Brezis and L. Nirenberg, 1983] and [M. Struwe, 1984] the Laplacian case p = 2 Similar nonlinearities in Conformal Geometry (Yamabe problem) and in Quantum Physics (Yang Mills equations) After > 25 years Theorem (C.M.- M. Willem, 2010) Let 1 < p < N and let u : R N + 0 be nonnegative solution of the equation zero on boundary. Then u 0. p u = µu p 1 in R N +, As a consequence precise description of those c s.t. (PS) c condition fails.

53 Critical points of zero on boundary. φ(u) = Ω u p p + a u p p µ u p p dx

54 Critical points of zero on boundary. Introduce φ(u) = Ω u p p RN u p φ (u) = p + a u p p µ u p p dx µ up + p dx.

55 Critical points of zero on boundary. Introduce Pick φ(u) = Ω u p p RN u p φ (u) = p (u n ) with small negative part. + a u p p µ u p p dx µ up + p dx. φ(u n ) c φ (u n ) 0,

56 We use the invariance under transformations T : z (λ) (p N)/p z(( y)/λ), λ R, y R N.

57 We use the invariance under transformations T : z (λ) (p N)/p z(( y)/λ), λ R, y R N. Theorem (C.M.-M. Willem, 2010) Passing to a subseq., there exist v 0 critical for φ and {v 1,..., v k } critical for φ s.t. u n v 0 k (λ i n) (p N)/p v i (( y i n)/λ i n) 0, n, i=1 k φ(v 0 ) + φ (v i ) = c. i=1 for some {y i n} n Ω and {λ i n} n R +,

58 Corollary (PS) c condition holds for those c such that φ admits only trivial critical points.

59 Corollary (PS) c condition holds for those c such that φ admits only trivial critical points. Consequence: sufficient conditions for the existence of solutions

60 Pure critical nonlinearity: { p u = u p 2 u in Ω u = 0 on Ω where Ω is annular shaped, i.e. 0 Ω and contains A R1,R 2 := {x R N : R 1 < x < R 2 }, invariant under the action of a closed subgroup of orthogonal transformations of R N.

61 Pure critical nonlinearity: { p u = u p 2 u in Ω u = 0 on Ω where Ω is annular shaped, i.e. 0 Ω and contains A R1,R 2 := {x R N : R 1 < x < R 2 }, invariant under the action of a closed subgroup of orthogonal transformations of R N. In [C.M.-F. Pacella, in progress] we prove existence of a positive solution.

62 Translation invariance in unbounded domains Problems but with translation invariance studied in [C.M.- M.Squassina, MM 2012] with diff. operators div(m ξ (u, Du)) + M s (u, Du)

63 Translation invariance in unbounded domains Problems but with translation invariance studied in [C.M.- M.Squassina, MM 2012] with diff. operators div(m ξ (u, Du)) + M s (u, Du) The Laplacian case studied in [V. Benci and G. Cerami, 1987]

64 Unbounded domains and unbounded PS sequences In the papers

65 Unbounded domains and unbounded PS sequences In the papers C. M., RL 2008

66 Unbounded domains and unbounded PS sequences In the papers C. M., RL 2008 D. Bonheure and C. M., JDE 2011

67 Unbounded domains and unbounded PS sequences In the papers C. M., RL 2008 D. Bonheure and C. M., JDE 2011 D. Bonheure, J. Di Cosmo and C. M., CCM 2012

68 Unbounded domains and unbounded PS sequences In the papers C. M., RL 2008 D. Bonheure and C. M., JDE 2011 D. Bonheure, J. Di Cosmo and C. M., CCM 2012 we studied the nonlinear Schrödinger-Poisson system (NSPS) in R N : u + V(x)u + ρ(x)φu = K(x)u p, φ = ρ(x)u 2.

69 Motivations R3 ρ(y) 2 ψ k + (V(x) E k )ψ k + ψ k (x) x y dy N ψ j (y)ψ k (y) ψ j (x) dy = 0 x y j=1 where ψ k : R 3 C "orthogonal", ρ := (1/N) N j=1 ψ j 2, V(x) ext. pot. E k "eigenvalues" (Lagrange-multipliers). Known as Hartree-Fock eq s for N electrons). R 3

70 Mean field approx. ρ u 2 where u s.t. R3 u(y) 2 u + V(x)u + Bu x y dy = C u 2/3 u, Schrödinger-Poisson-Slater system (see [Bokanowski-Lopez-Soler, 2003] or [Mauser, 2001]).

71 Mean field approx. ρ u 2 where u s.t. R3 u(y) 2 u + V(x)u + Bu x y dy = C u 2/3 u, Schrödinger-Poisson-Slater system (see [Bokanowski-Lopez-Soler, 2003] or [Mauser, 2001]). NSPS is a generalization. We find existence, non-existence, multiplicity and qualitative behavior of critical points of J(u) := 1 ( u 2 + V(x)u 2 )dx + 1 φ u u 2 dx 2 R N 4 R N 1 K(x) u p+1 dx. p + 1 R N

72 Difficulties:

73 Difficulties: Domain is R N

74 Difficulties: Domain is R N Difficult to prove (PS) c seq. s are bounded (open prob.)

75 Difficulties: Domain is R N Difficult to prove (PS) c seq. s are bounded (open prob.) With Mark, Kohn-Sham eq s for deformed cristals or Kármán-Donnell eq s involving the biharmonic operator, present similar difficulties.

76