Positive constrained minimizers for supercritical problems in the ball

TL;DR

This paper establishes conditions for the existence of positive solutions to a supercritical nonlinear PDE in a ball, using constrained minimizers of the energy functional, applicable to both Neumann and Dirichlet boundary conditions.

Contribution

It provides a new sufficient condition for positive solutions in supercritical problems, extending previous results to large p and both boundary conditions.

Findings

Existence of positive solutions for large p in the supercritical regime.

Solutions are constrained minimizers of the energy functional.

Neumann problem admits solutions under mild conditions on V.

Abstract

We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns to be a constrained minimum of the associated energy functional. As an application we show that, in case $V(|x|)$ is smooth, nonnegative and not identically zero, and p is sufficiently large, the Neumann problem always admits a solution.