A $p$-Laplacian Neumann problem with a possibly supercritical nonlinearity

Abstract

We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-{\Delta }_{p}u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of ${\mathbb{R}}^N$, subject to homogeneous Neumann boundary conditions. The assumptions on the nonlinearity $f$ are very mild and allow it to be possibly supercritical in the sense of Sobolev embeddings. The main tools used are the truncation method and a mountain pass-type argument. In the pure power case, i.e., $f(u)=u^{q-1}$, we detect the limit profile of the solutions of the problems as $q\to \infty$.

9 pages, 2 figures, BRU-TO PDE's Conference

Cited by in corpus (3)

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