Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains

Abstract

Given $\rho >0$, we study the elliptic problem

$$\text{ find }\left(U,\lambda \right)\in H_0^1\left(B_1\right)\times ℝ\text{ such that }\left\{\begin{array}{c}\hfill \\ \hfill \\ -\Delta U+\lambda U=U^p,\hfill \\ {\int }_{B_1}{U}^{2}dx=\rho ,\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.U>0,$$

where $B_1\subset {ℝ}^N$ is the unitary ball and $p$ is Sobolev-subcritical. Such a problem arises in the search for solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about $\rho$, $N$ and $p$) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every $\rho$ (in the existence range) when $p$ is $L^2$-critical and subcritical, i.e., $1<p\le 1+4∕N$, while they are stable for almost every $\rho$ in the $L^2$-supercritical regime $1+4∕N<p<2^{\ast }-1$. The proofs are obtained in connection with the study of a variational problem with two constraints of independent interest: to maximize the $L^{p+1}$-norm among functions having prescribed $L^2$- and $H_0^1$-norms.