Positive constrained minimizers for supercritical problems in the ball (Q2884434)

This paper studies the existence of radial solutions to the equation \(\Delta u+V(| x|)u=u^p\) in the unit ball of \(\mathbb R^n\), \(n\geq 2\). The authors provide a sufficient condition for the existence of a positive solution to the equation above, when \(p\) is large enough both with Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application, they show that the Neumann problem always admits a solution if \(V(| x|)\geq 0\), \(V\not\equiv 0\) is smooth and \(p\) is sufficiently large.