Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case

Abstract

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities \[ -Δu= λu + μ|u|^{q-2} u + |u|^{2^*-2} u \qquad \text{in ${\mathbb{R}}^N$, $N\ge 3$,} \] having prescribed mass \[ \int_{\mathbb{R}^N} |u|^2 = a^2, \] in the \emph{Sobolev critical case}. For a $L^2$-subcritical, $L^2$-critical, of $L^2$-supercritical perturbation $\hat{I}¼|u{|}^{q-2}u$ we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space ${\mathbb{R}}^N$.

arXiv admin note: text overlap with arXiv:1811.00826 Final version, accepted on Journal of Functional Analysis