Positive constrained minimizers for supercritical problems in the ball

We provide a sufficient condition for the existence of a positive solution to

$$-\Delta u+V(\vert x\vert) u=u^p \quad \hbox { in } B_1,$$

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 $V(\vert x\vert)\geq 0$, $V\not \equiv 0$ is smooth and $p$ is sufficiently large, and the Neumann problem always admits a solution.