Potential systems with singular $\hat{I}¦$-Laplacian

Abstract

We are concerned with solvability of the boundary value problem $$-{\left[\ddot{I}•(u^{\prime })\right]}^{\prime }={\nabla }_{u}F(t,u),\left(\ddot{I}•\left(u^{\prime }\right)(0),-\ddot{I}•\left(u^{\prime }\right)(T)\right)\in \partial j(u(0),u(T)),$$ where $\ddot{I}•$ is a homeomorphism from $B_a$ -- the open ball of radius $a$ centered at $0_{{\mathbb{R}}^N},$ onto ${\mathbb{R}}^N$, satisfying $\ddot{I}•(0_{{\mathbb{R}}^N})=0_{{\mathbb{R}}^N}$, $\ddot{I}•=\nabla \hat{I}¦$, with $\hat{I}¦:{\overline{B}}_a\to (-\infty ,0]$ of class $C^1$ on $B_a$, continuous and strictly convex on ${\overline{B}}_a.$ The potential $F:[0,T]\times {\mathbb{R}}^N\to \mathbb{R}$ is of class $C^1$ with respect to the second variable and $j:{\mathbb{R}}^N\times {\mathbb{R}}^N\to (-\infty ,+\infty ]$ is proper, convex and lower semicontinuous. We first provide a variational formulation in the frame of critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals. Then, taking the advantage of this key step, we obtain existence of minimum energy as well as saddle-point solutions of the problem. Some concrete illustrative examples of applications are provided.