Nontrivial solutions to affine p-Laplace equations via a perturbation strategy

Abstract

This paper is concerned with the existence of nontrivial solutions for affine $p$-Laplace equations involving subcritical nonlinearities behaving at $u=0$ as $u^q$ with $q<p-1$ and at the infinity as $u^r$ with $r>p-1$. Since local Palais–Smale compactness for affine energy type functionals is an open hard question, the problem is overcome by means of a perturbative approach using the space norm. Mountain-pass critical points are constructed from a limit process of corresponding ones in the modified affine context. Compactness and stability of MP solution sets are also addressed.