Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case (1901.02003v1)
Abstract: We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities [ -\Delta u= \lambda u + \mu |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 = a2, ] in the \emph{Sobolev critical case}. For a $L2$-subcritical, $L2$-critical, of $L2$-supercritical perturbation $\mu |u|{q-2} u$ 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 $\mathbb{R}N$.