Potential systems with singular $\mathrm{\Phi }$-Laplacian

We are concerned with solvability of the boundary value problem $$-{[\phi (u^{\mathrm{\prime }})]}^{\mathrm{\prime }}={\mathrm{\nabla }}_{u}F(t,u),(\phi \left(u^{\mathrm{\prime }}\right)(0),-\phi \left(u^{\mathrm{\prime }}\right)(T))\in \mathrm{\partial }j(u(0),u(T)),$$ 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 $\phi (0_{{\mathbb{R}}^N})=0_{{\mathbb{R}}^N}$, $\phi =\mathrm{\nabla }\mathrm{\Phi }$, with $\mathrm{\Phi }:{\bar{B}}_a\to (-\mathrm{\infty },0]$ of class $C^1$ on $B_a$, continuous and strictly convex on ${\bar{B}}_a\mathrm{.}$ The potential $F:[0,T]\times {\mathbb{R}}^N\to \mathbb{R}$ is of class $C^1$ with respect to the second variable and $j:{\mathbb{R}}^N\times {\mathbb{R}}^N\to (-\mathrm{\infty },+\mathrm{\infty }]$ 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.