Potential systems with singular -Laplacian

We are concerned with solvability of the boundary value problem - (u^) ^=ᵤ F (t, u), ( (u^) (0), - (u^) (T) ) j (u (0), u (T) ), where is a homeomorphism from Bₐ -- the open ball of radius a centered at 0ₑ₍, onto RN, satisfying (0ₑ₍) =0ₑ₍, =, with: Bₐ (-, 0] of class C¹ on Bₐ, continuous and strictly convex on Bₐ. 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.