Solutions with prescribed mass for nonlinear Schrödinger equations

Transcription

1 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 Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

2 The problem Let R N be a bounded domain, 1 < p < 2 1, ρ > 0 be given. Find (U, λ) H0 1 () R s.t. Main goals: existence and multiplicity results; U + λu = U p 1 U U 2 dx = ρ stability results for ground states (to be defined later). Warning : two critical exponents, p = 2 1 = (Sobolev crit. exp.), N 2 p = N (L2 crit. exp.). We understand (super)criticality only in the L 2 sense. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

3 Motivations Standing wave solutions of the focusing nonlinear Schrödinger equation (NLS) { i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R Φ(t, x) = 0 (t, x) R, Φ(t, x) = eiλt U(x) (U real valued) U + λu = U p 1 U, U H 1 0 () ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

4 Motivations Standing wave solutions of the focusing nonlinear Schrödinger equation (NLS) { i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R Φ(t, x) = 0 (t, x) R, Φ(t, x) = eiλt U(x) (U real valued) U + λu = U p 1 U, U H 1 0 () NLS on bounded domains appears in different physical contexts: Nonlinear optics (N = 2, p = 3, = disk): propagation of laser beams in hollow core fibers. [Fibich, Merle (2001)] Bose Einstein condensation (N 3, p = 3): confined particles in quantum mechanical systems in the presence of an infinite well trapping potential [Lieb et al (2006), Bartsch, Parnet (2012)] ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

5 Basic facts about NLS i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R Φ(t, x) = 0 (t, x) R, Φ(0, x) = Φ 0 (x) x, Conserved quantities along trajectories (at least formally): ( 1 Energy: E(Φ) = 2 Φ 2 1 ) p + 1 Φ p+1 dx Mass (or Charge): Q(Φ) = Φ 2 dx. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

6 Basic facts about NLS i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R Φ(t, x) = 0 (t, x) R, Φ(0, x) = Φ 0 (x) x, Conserved quantities along trajectories (at least formally): ( 1 Energy: E(Φ) = 2 Φ 2 1 ) p + 1 Φ p+1 dx Mass (or Charge): Q(Φ) = Φ 2 dx. Orbital stability The standing wave e iλt U(x) is orbitally stable if ε > 0 δ > 0 such that: Φ 0 U H 1 0 (,C) < δ = sup 0<t< inf Φ(t, ) s R eiλs U H 1 0 (,C) < ε, where Φ(t, x) is the solution of (NLS) with Φ(0, ) = Φ 0, which is required to enjoy global existence as well. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

7 The L 2 -critical exponent in R N i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R R N Subcritical Case (1 < p < 1 + 4/N): global existence for all initial data. Critical Case (p = 1 + 4/N): global existence for data with small mass Q. Supercritical Case (p > 1 + 4/N): explosion in finite time. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

8 The L 2 -critical exponent in R N i Φ t + Φ + Φ p 1 Φ = 0 (t, x) R R N Subcritical Case (1 < p < 1 + 4/N): global existence for all initial data. Critical Case (p = 1 + 4/N): global existence for data with small mass Q. Supercritical Case (p > 1 + 4/N): explosion in finite time. Let Z N,p be the unique solution (up to translations) of Z + Z = Z p, Z H 1 (R N ), Z > 0. Then e it Z N,p is orbitally stable if 1 + 4/N (subcritical); e it Z N,p is unstable if p 1 + 4/N (critical and supercritical). Proofs: [Coffman (1972), Kwong ARMA (1989)], [Cazenave, Lions CMP (1982)]. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

9 Two points of view about standing waves... Going back to the elliptic problem: U + λu = U p 1 U, U H 1 0 () The energy and mass are respectively: E(U) = 1 U 2 dx 1 U p+1 dx, Q(U) = 2 p + 1 Two points of view: 1 Chemical Potential λ R given Solutions are critical points of the Action Functional: A λ (U) = E(U) + λ 2 Q(U) U 2 dx ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

10 Two points of view about standing waves... Going back to the elliptic problem: U + λu = U p 1 U, U H 1 0 () The energy and mass are respectively: E(U) = 1 U 2 dx 1 U p+1 dx, Q(U) = 2 p + 1 Two points of view: 1 Chemical Potential λ R given Solutions are critical points of the Action Functional: A λ (U) = E(U) + λ 2 Q(U) 2 The mass Q(U) = ρ is fixed and λ is an unknown of the problem Find critical points of (λ appears as a Lagrange multiplier) E {Q=ρ} U 2 dx ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

11 ...and two notions of ground states 1 Least Action Solutions (λ given) : minimize the action A λ among its nontrivial critical points. a λ = inf{a λ (U) : U H 1 0, U 0, A λ(u) = 0} [Berestycki, Lions ARMA (1983)] As an equivalent definition, one can minimize on the (smooth) Nehari manifold. Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

12 ...and two notions of ground states 1 Least Action Solutions (λ given) : minimize the action A λ among its nontrivial critical points. a λ = inf{a λ (U) : U H 1 0, U 0, A λ(u) = 0} [Berestycki, Lions ARMA (1983)] As an equivalent definition, one can minimize on the (smooth) Nehari manifold. 2 Least Energy Solutions (λ unknown, mass fixed) : [Cazenave, Lions CMP (1982)] inf E(U) Q(U)=ρ ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

13 ...and two notions of ground states 1 Least Action Solutions (λ given) : minimize the action A λ among its nontrivial critical points. a λ = inf{a λ (U) : U H 1 0, U 0, A λ(u) = 0} [Berestycki, Lions ARMA (1983)] As an equivalent definition, one can minimize on the (smooth) Nehari manifold. 2 Least Energy Solutions (λ unknown, mass fixed) : [Cazenave, Lions CMP (1982)] inf E(U) Q(U)=ρ Problem: The infimum is equal to in the supercritical case p > 1 + 4/N, and in the critical case p = 1 + 4/N for large ρ (Gagliardo Nirenberg). ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

14 Least Energy Solutions Definition For ρ > 0, denote the set of positive solutions with prescribed mass ρ by: } P ρ = {U H0 1 () : Q(U) = ρ, U > 0, E {Q=ρ} (U) = 0 = { U H 1 0 () : Q(U) = ρ, U > 0, λ : U + λu = U p} A positive least energy solution is a minimizer of the problem e ρ = inf P ρ E. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

15 Least Energy Solutions Definition For ρ > 0, denote the set of positive solutions with prescribed mass ρ by: } P ρ = {U H0 1 () : Q(U) = ρ, U > 0, E {Q=ρ} (U) = 0 = { U H 1 0 () : Q(U) = ρ, U > 0, λ : U + λu = U p} A positive least energy solution is a minimizer of the problem e ρ = inf P ρ E. When = R N, both the problems can be uniquely solved by suitable scaling of Z N,p solution of Z + Z = Z p, Z H 1 (R N ), Z > 0. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

16 Least Energy Solutions Definition For ρ > 0, denote the set of positive solutions with prescribed mass ρ by: } P ρ = {U H0 1 () : Q(U) = ρ, U > 0, E {Q=ρ} (U) = 0 = { U H 1 0 () : Q(U) = ρ, U > 0, λ : U + λu = U p} A positive least energy solution is a minimizer of the problem e ρ = inf P ρ E. When = R N, both the problems can be uniquely solved by suitable scaling of Z N,p solution of Z + Z = Z p, Z H 1 (R N ), Z > 0. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

17 Existence of positive solutions in B 1 (0) Theorem 1 If 1 < p < 1 + 4/N then, for every ρ > 0, there exists a unique positive solution, which achieves e ρ ; ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

18 Existence of positive solutions in B 1 (0) Theorem 1 If 1 < p < 1 + 4/N then, for every ρ > 0, there exists a unique positive solution, which achieves e ρ ; 2 if p = 1 + 4/N, for 0 < ρ < Z N,p 2 L 2 (R N ), there exists a unique positive solution, which achieves e ρ; for ρ Z N,p 2 L 2 (R N ), no positive solution exists; ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

19 Existence of positive solutions in B 1 (0) Theorem 1 If 1 < p < 1 + 4/N then, for every ρ > 0, there exists a unique positive solution, which achieves e ρ ; 2 if p = 1 + 4/N, for 0 < ρ < Z N,p 2 L 2 (R N ), there exists a unique positive solution, which achieves e ρ; for ρ Z N,p 2 L 2 (R N ), no positive solution exists; 3 if 1 + 4/N < p < 2 1, there exists ρ > 0 such that: e ρ is achieved if and only if 0 < ρ ρ. no positive solutions exists for ρ > ρ, For 0 < ρ < ρ there exist at least two distinct positive solutions. In this latter case, P ρ contains positive solutions of the problem which are not least energy solutions. [B.Noris, H.Tavares and G. Verzini (APDE 2014)] ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

20 Existence of positive solutions in B 1 (0) Theorem 1 If 1 < p < 1 + 4/N then, for every ρ > 0, there exists a unique positive solution, which achieves e ρ ; 2 if p = 1 + 4/N, for 0 < ρ < Z N,p 2 L 2 (R N ), there exists a unique positive solution, which achieves e ρ; for ρ Z N,p 2 L 2 (R N ), no positive solution exists; 3 if 1 + 4/N < p < 2 1, there exists ρ > 0 such that: e ρ is achieved if and only if 0 < ρ ρ. no positive solutions exists for ρ > ρ, For 0 < ρ < ρ there exist at least two distinct positive solutions. In this latter case, P ρ contains positive solutions of the problem which are not least energy solutions. [B.Noris, H.Tavares and G. Verzini (APDE 2014)] Remark. For λ > λ 1 (B 1 ) there is a unique (positive) least action solution; then, if p > 1 + 4/N there are least action solutions which are not least energy solutions. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

21 Orbital stability Let U denote a least energy solution of U + λu = U p 1 U, U H0 1(B 1) U 2 dx = ρ, U > 0. B 1 and let Φ(t, x) = e iλt U(x). Theorem If 1 < p 1 + 4/N (subcritical and critical) then Φ is orbitally stable; if 1 + 4/N < p < 2 1 (supercritical) then Φ is orbitally stable for a.e. ρ (0, ρ ]. [B.Noris, H.Tavares and G. Verzini (APDE, 2014)] Remark: comparison with the results of the problem in R n suggests that the presence of a boundary has a stabilizing effect [Fibich, Merle Phys. D (2001)] ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

22 Existence by constrained optimization Change of variable: Let R N be any bounded domain U + λu = U p 1 U U 2 dx = ρ U(x)= ρu(x) µ=ρ (p 1)/2 u + λu = µ u p 1 u u 2 dx = 1 ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

23 Existence by constrained optimization Change of variable: Let R N be any bounded domain U + λu = U p 1 U U 2 dx = ρ U(x)= ρu(x) µ=ρ (p 1)/2 u + λu = µ u p 1 u u 2 dx = 1 Optimization with two constraints: For each α > λ 1 (), define { U α := u H0 1 () : and u 2 dx = 1, M α = sup u p+1 dx u U α } u 2 dx = α. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

24 M α is related to Gagliardo-Nirenberg inequality: u p+1 L p+1 () C N,p u p+1 N(p 1)/2 L 2 () u N(p 1)/2 L 2 (), u H 1 0 () by: C N,p = sup α λ 1 () M α α N(p 1)/2. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

25 M α is related to Gagliardo-Nirenberg inequality: u p+1 L p+1 () C N,p u p+1 N(p 1)/2 L 2 () u N(p 1)/2 L 2 (), u H 1 0 () by: C N,p = sup α λ 1 () M α α N(p 1)/2. Theorem M α is achieved by a positive function u H0 1 (), and there exist µ > 0, λ > λ 1 () such that u + λu = µu p, u 2 dx = 1, u 2 dx = α ( ) Moreover, for each small ε > 0 and α (λ 1, λ 1 + ε) there is exactly one solution u of ( ) with λ > λ 1, µ > 0 and s.t. up+1 = M α. [B.Noris, H.Tavares and G. Verzini (APDE 2014)] ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

26 The case = B 1 (0) Combining results/ideas of many authors [Gidas Ni Nirenberg, Kwong, Kwong Li, Korman, Aftalion Pacella, Felmer Martínez Tanaka] we have uniqueness of positive solutions, which are nondegenerate and with Morse index always one. As a consequence, the set { } (u, µ, λ) : u + λu = µu p, u > 0, u 2 = 1, µ > 0 B 1 can be parameterized with α in a smooth way: α (u(α), µ(α), λ(α)), where B 1 u(α) 2 = α, and u(α) achieves M α. Moreover, λ (α) > 0. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

27 The case = B 1 (0) Combining results/ideas of many authors [Gidas Ni Nirenberg, Kwong, Kwong Li, Korman, Aftalion Pacella, Felmer Martínez Tanaka] we have uniqueness of positive solutions, which are nondegenerate and with Morse index always one. As a consequence, the set { } (u, µ, λ) : u + λu = µu p, u > 0, u 2 = 1, µ > 0 B 1 can be parameterized with α in a smooth way: α (u(α), µ(α), λ(α)), where B 1 u(α) 2 = α, and u(α) achieves M α. Moreover, λ (α) > 0. The behavior of α µ(α) is crucial; in fact: prescribing the mass ρ is equivalent to prescribing µ = ρ (p 1)/2 ; µ positive (resp. negative) implies orbital stability (resp. instability) of the corresponding standing waves [Grillakis, Shatah, Strauss JFA (1987)]. ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

28 For p N we have µ > 0 for every α. Existence and stability of positive solutions follow for any given mass ρ > 0 if p is subcritical and for 0 < ρ < Z N,p 2 in the critical case. L 2 (R N ) Supercritical case (p > 1 + 4/N) : lim α + µ(α) = Numerical simulation for N = 3, p = 3 At least two solutions for 0 < ρ < ρ, stability of least energy solutions. Α Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

29 An estimate of ρ (in a general bounded domain) One constraint optimization : Recall the definition { } U α := u H0 1 () : u 2 dx = 1, u 2 dx = α and define further: B α := For any given µ > 0, let Theorem { } u H0 1 () : u 2 dx = 1, u 2 dx < α E µ (u) := 1 2 u 2 dx µ u p+1 dx p + 1 Let α > λ 1 () and assume that E Uα µ > E µ (φ 1 ) (φ 1 being the first normalized Dirichlet eigenfunction). Then, inf u Bα E µ (u) is a critical value of E µ on the manifold M := {u H0 1(), u L 2 () = 1}. [G.Verzini, D.P.] ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

30 By Gagliardo-Nirenberg, for u U α we have E µ (u) > 1 2 α µ p + 1 C N,p α N(p 1)/4 Hence, the assumption of the theorem holds for 0 < µ p α λ 1 () C N,p α N(p 1)/4 φ 1 p+1 L p+1 () f (α) For p > N, the function f (α) tends to zero for α + and assume a maximum µ depending on N, p and. Thus: Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

31 For 0 < µ µ (and for α in a suitable interval) there is a constrained minimum point u µ B α (u µ > 0) s.t. u µ + λ µ u µ = µu p µ for some λ µ R; moreover, u µ 2 = α µ < α. u µ is also the minimum point of E µ on U αµ, and therefore is the maximum of u p+1 on the same set due to E Uαµ µ = 1 2 α µ µ u p+1 dx p + 1 It follows that there are least energy solutions of mass ρ (0, ρ ] where ρ = (µ ) 2 p 1 Remark: In the critical case, we recover ρ (0, Z N,p 2 L 2 (R N ) ). In the subcritical case, E µ is bounded below and coercive on the manifold M for every µ, hence ρ = +. Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

32 Changing-sign solutions For p critical and supercritical and ρ large no positive solution exists. What about changing-sign ones? ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

33 Changing-sign solutions For p critical and supercritical and ρ large no positive solution exists. What about changing-sign ones? Back to the two constraints : { U α := u H0 1 () : Theorem If λ k < α < λ k+1 then and, for 1 h k, M α,h = u 2 dx = 1, genus(u α ) = k, max A Uα genus(a) h min u p+1 dx u A } u 2 dx = α. is achieved by some u U α, with multipliers µ > 0, λ > λ k+1, such that [G.Verzini, D.P.] u + λu = µ u p 1 u ario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

34 Asymptotics for α + If u n is such that u n + λ n u = µ n u n p 1 u n with u n 2 = α n + and with finite Morse index, then λ n +. If x n is a local maximum for u n, v n (x) := ( µn λ n ) 1/(p 1) u n ( x λn + x n ) satisfies, up to subs., v n W in C 1 loc(r N ) where W solves W + W = W p 1 W, in H 1 (R N ) W = Z N,p for positive solutions [Druet, Hebey and Robert; Esposito and Petralla]. Conjecture: in the critical case, there is v n and W s.t. µ n W p 1 L 2 (R N ) > 2 p 1 2 Z N,p p 1 L 2 (R N ) Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20

35 Thank you! Dario Pierotti (Dipartimento di Matematica, Politecnico Solutions di Milano (ITALY)) with prescribed mass for nonlinear Schrödinger equationsvarese - September 17, / 20