## Existence and orbital stability of the ground states with prescribed mass for the $$L^2$$-critical and supercritical NLS on bounded domains.(English)Zbl 1314.35168

Summary: Given $$\rho>0$$, we study the elliptic problem $\text{find $$(U,\lambda)\in H^1_0(B_1)\times \mathbb{R}$$ such that $$\begin{cases} -\Delta U+\lambda U=U^p, \\ \int_{B_1} U^2\, dx=\rho, \end{cases}$$} U>0,$ where $$B_1\subset\mathbb{R}^N$$ is the unitary ball and $$p$$ is Sobolev-subcritical. Such a problem arises in the search for solitary wave solutions for nonlinear Schr\'’odinger 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., $$1<p\leq1+4/N$$, 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 $$H^1_0$$-norms.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35J20 Variational methods for second-order elliptic equations 35C08 Soliton solutions
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