Positive constrained minimizers for supercritical problems in the ball

Abstract:

We provide a sufficient condition for the existence of a positive solution to \[ -\Delta u+V(|x|) u=u^p \quad \hbox { in } B_1, \] when $p$ is large enough. Here $B_1$ is the unit ball of $\mathbb {R}^n$, $n\ge 2$, and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case $V(|x|)\geq 0$, $V\not \equiv 0$ is smooth and $p$ is sufficiently large, and the Neumann problem always admits a solution.