Variational methods for nonlinear perturbations of singular φ-Laplacians

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space div(del v/root 1 - vertical bar del v vertical bar(2)) = g(vertical bar x vertical bar, v) in A, partial derivative v/partial derivative v = 0 on partial derivative A, where 0 <= R(1) < R(2), A = {x is an element of R(N): R(1) <= vertical bar x vertical bar <= R(2)} and g : [R(1); R(2)] x R -> R is continuous, we study the more general problem [r(N-1)phi(u')]' = r(N-1)g(r, u), u'(R(1)) = 0 = u'(R(2)), where phi := Phi': [-a, a] -> R is an increasing homeomorphism with phi(0) = 0 and the continuous function Phi : [-a, a] -> R is of class C(1) on (-a, a). The associated functional in the space of continuous functions over [R(1), R(2)] is the sum of a convex lower semicontinuous functional and of a functional of class C(1). Using the critical point theory of Szulkin, we obtain various existence and multiplicity results for several classes of nonlinearities. We also discuss the case of the periodic problem.