JournalsrlmVol. 22, No. 1pp. 89–111

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

• ### Petru Jebelean

West University of Timisoara, Romania
• ### Jean Mawhin

Université Catholique de Louvain, Belgium 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 $0 \leq R_1 < R_2$, $\mathcal A=\{ x\in \mathbb{R}^N: R_1\leq |x|\leq R_2\}$ and $g:[R_1,R_2]\times \mathbb{R}\to\mathbb{R}$ 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 $\phi:=\Phi':(-a,a) \to \mathbb{R}$ is an increasing homeomorphism with $\phi(0)=0$ 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.