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

Noris, Benedetta; Tavares, Hugo; Verzini, Gianmaria

Anal. PDE 7, No. 8, 1807-1838 (2014).

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

Keywords:

Gagliardo-Nirenberg inequality; constrained critical points; Ambrosetti-Prodi-type problem; singular perturbations