Radial positive solutions for p-Laplacian supercritical Neumann problems

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a supercritical p-Laplacian Neumann problem in a ball, using variational and shooting methods, revealing multiple solutions even with supercritical nonlinearities.

Contribution

It introduces new results on positive solutions for supercritical p-Laplacian problems with Neumann boundary conditions, combining variational and shooting techniques.

Findings

Existence of at least one non-zero constant solution

Multiple positive solutions under supercritical conditions

Application of variational and shooting methods to nonlinear PDEs

Abstract

This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in [6,3].