Existence of Solutions for a Non-homogeneous Neumann Problem

Abstract

The aim of this paper is to study the following non-homogeneous Neumann-type problem

$$\begin{aligned} \left\{ \begin{array} {ll} - \mathrm{{div}}(\alpha (\vert \nabla u\vert )\nabla u) + u = u \vert u\vert ^{p - 2}, &{} \quad \mathrm{in } \, B_{1}, \\ \dfrac{\partial u}{\partial \nu } = 0, &{} \quad \mathrm{on } \, \partial B_{1} , \end{array}\right. \end{aligned}$$

(0.1)

where $B_{1}$ is the unit ball in $\mathbb {R}^{n}$ and $p > 2$. We establish the existence of a non-constant, positive, radially non-decreasing weak solution for (0.1), under certain assumptions on $\alpha$. Our approach relies on the theory of Orlicz spaces combined with a new variational method that allows one to deal with problems beyond the usual locally compactness structure and a variant of Mountain Pass Theorem.