Potential systems with singular $\Phi$-Laplacian

We are concerned with solvability of the boundary value problem $$-\left[ \phi(u^{\prime}) \right] ^{\prime}=\nabla_u F(t,u), \quad \left ( \phi \left( u^{\prime }\right)(0), -\phi \left( u^{\prime }\right)(T)\right )\in \partial j(u(0), u(T)),$$ where $\phi$ is a homeomorphism from $B_a$ -- the open ball of radius $a$ centered at $0_{\mathbb{R}^N},$ onto $\mathbb{R}^N$, satisfying $\phi(0_{\mathbb{R}^N})=0_{\mathbb{R}^N}$, $\phi =\nabla \Phi$, with $\Phi: \overline{B}_a \to (-\infty, 0]$ 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 $j:\mathbb{R}^N \times \mathbb{R}^N \rightarrow (-\infty, +\infty]$ 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.