Variational methods for nonlinear perturbations of singular $\varphi$-Laplacians

Abstract

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space

where $0\le R_1<R_2$, $\mathcal{A}=\{x\in {\mathbb{R}}^N:R_1\le |x|\le R_2\}$ and $g:[R_1,R_2]\times \mathbb{R}\to \mathbb{R}$ is continuous, we study the more general problem

where $\varphi :={\Phi }^{\prime }:(-a,a)\to \mathbb{R}$ is an increasing homeomorphism with $\varphi (0)=0$ and the continuous function $\Phi :[-a,a]\to \mathbb{R}$ is of class $C^1$ on $(-a,a)$. The associated functional in the space of continuous functions over $[R_1,R_2]$ is the sum of a convex lower semicontinuous functional and of a functional of class $C^1$. Using the critical point theory of Szulkin, we obtain various existence and multiplicity results for several classes of nonlinearities. We also discuss the case of the periodic problem.