#### Vol. 7, No. 8, 2014

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

### Benedetta Noris, Hugo Tavares and Gianmaria Verzini

Vol. 7 (2014), No. 8, 1807–1838
##### Abstract

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

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, while they are stable for almost every $\rho$ in the ${L}^{2}$-supercritical regime $1+4∕N. 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.

##### Keywords
Gagliardo–Nirenberg inequality, constrained critical points, Ambrosetti–Prodi-type problem, singular perturbations
##### Mathematical Subject Classification 2010
Primary: 35B35, 35J20, 35Q55, 35C08