A p-Laplacian supercritical Neumann problem

Abstract

For $p>2$, we consider the quasilinear equation $-{\Delta }_{p}u+|u{|}^{p-2}u=g(u)$ in the unit ball $B$ of ${\mathbb{R}}^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u{|}^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to \infty$.

34 pages, 1 figure

References in corpus (1)

• Highly oscillatory solutions of a Neumann problem for a $p$-laplacian equation

Cited by in corpus (2)

• A priori bounds and multiplicity of positive solutions for $p$-Laplacian Neumann problems with sub-critical growth
• A nonlocal supercritical Neumann problem