# Variational methods for nonlinear perturbations of singular <i>ϕ</i>-Laplacians

### Cristian Bereanu

Romanian Academy, Bucharest, Romania### Petru Jebelean

West University of Timisoara, Romania### Jean Mawhin

Université Catholique de Louvain, Belgium

## Abstract

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space \begin{eqnarray*} \mbox{div}\left(\frac{\nabla v}{\sqrt{1-|\nabla v|^2}}\right)=g(|x|,v) \quad \mbox{in} \quad \mathcal A, \quad \frac{\partial v}{\partial \nu}=0 \quad \mbox{on} \quad \partial\mathcal A, \end{eqnarray*} where $0≤R_{1}<R_{2}$, $A={x∈R_{N}:R_{1}≤∣x∣≤R_{2}}$ and $g:[R_{1},R_{2}]×R→R$ is continuous, we study the more general problem \begin{eqnarray*} [r^{N-1} \phi(u')]' =r^{N-1}g(r,u), \quad u'(R_1)=0=u'(R_2), \end{eqnarray*} where $ϕ:=Φ_{′}:(−a,a)→R$ is an increasing homeomorphism with $ϕ(0)=0$ and the continuous function $Φ:[−a,a]→R$ 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.

## Cite this article

Cristian Bereanu, Petru Jebelean, Jean Mawhin, Variational methods for nonlinear perturbations of singular <i>ϕ</i>-Laplacians. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 1, pp. 89–111

DOI 10.4171/RLM/589