Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Abstract

Given $$\rho >0$$ , we study the elliptic problem $$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain and $$p>1$$ is Sobolev-subcritical, searching for conditions (about $$\rho $$ , N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is $$L^2$$ -subcritical, i.e. $$1<p<1+4/N$$ , the problem admits solutions for every $$\rho >0$$ . In the $$L^2$$ -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ , we show that, for any $$k\in {\mathbb {N}}$$ , the problem admits solutions having Morse index bounded above by k only if $$\rho $$ is sufficiently small. Next we provide existence results for certain ranges of $$\rho $$ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta $$ in $$H^1_0(\Omega )$$ , 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.