Potential systems with singular Φ-Laplacian

We are concerned with solvability of the boundary value problem -[ φ(u^) ] ^=∇ᵤ F(t,u), ( φ ( u)(0), -φ ( u)(T) )∈ ∂ j(u(0), u(T)), where φ is a homeomorphism from Bₐ -- the open ball of radius a centered at 0RN, onto RN, satisfying φ(0RN)=0RN, φ =∇ Φ, with Φ: B̄ₐ → (-∞, 0] of class C¹ on Bₐ, continuous and strictly convex on B̄ₐ. The potential F:[0,T] × RN → R is of class C¹ with respect to the second variable and j:RN × RN → (-∞, +∞] 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 C¹-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.