Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Normalized bound states for the nonlinear Schrödinger equation in bounded domains Given $$\rho >0$$ ρ > 0 , we study the elliptic problem \begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned} find ( U , λ ) ∈ H 0 1 ( Ω ) × R such that - Δ U + λ U = | U | p - 1 U ∫ Ω U 2 d x = ρ , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain and $$p>1$$ p > 1 is Sobolev-subcritical, searching for conditions (about $$\rho$$ ρ , N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is $$L^2$$ L 2 -subcritical, i.e. $$1<p<1+4/N$$ 1 < p < 1 + 4 / N , the problem admits solutions for every $$\rho >0$$ ρ > 0 . In the $$L^2$$ L 2 -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ 1 + 4 / N ≤ p < 2 ∗ - 1 , we show that, for any $$k\in {\mathbb {N}}$$ k ∈ N , the problem admits solutions having Morse index bounded above by k only if $$\rho$$ ρ is sufficiently small. Next we provide existence results for certain ranges of $$\rho$$ ρ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta$$ - Δ in $$H^1_0(\Omega )$$ H 0 1 ( Ω ) , extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

, Volume 56 (5) – Aug 23, 2017
27 pages

/lp/springer_journal/normalized-bound-states-for-the-nonlinear-schr-dinger-equation-in-8zp9ngCUFJ
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1232-7
Publisher site
See Article on Publisher Site

Abstract

Given $$\rho >0$$ ρ > 0 , we study the elliptic problem \begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned} find ( U , λ ) ∈ H 0 1 ( Ω ) × R such that - Δ U + λ U = | U | p - 1 U ∫ Ω U 2 d x = ρ , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain and $$p>1$$ p > 1 is Sobolev-subcritical, searching for conditions (about $$\rho$$ ρ , N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is $$L^2$$ L 2 -subcritical, i.e. $$1<p<1+4/N$$ 1 < p < 1 + 4 / N , the problem admits solutions for every $$\rho >0$$ ρ > 0 . In the $$L^2$$ L 2 -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ 1 + 4 / N ≤ p < 2 ∗ - 1 , we show that, for any $$k\in {\mathbb {N}}$$ k ∈ N , the problem admits solutions having Morse index bounded above by k only if $$\rho$$ ρ is sufficiently small. Next we provide existence results for certain ranges of $$\rho$$ ρ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta$$ - Δ in $$H^1_0(\Omega )$$ H 0 1 ( Ω ) , extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Aug 23, 2017

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