Variational methods for nonlinear perturbations of singular φ-Laplacians

Abstract

Motivated by the existence of radial solutions to the Neumann problem involving the mean extrinsic curvature operator in Minkowski space div(EQUATION PRESANT) where 0 ≤ R1 < R2, A = {x a RN: R1 ≤ |x| a R2} and g: [R1; R2]x R → R is continuous, we study the more general problem [rN- 1φ{u')]' = rN-1g(r;u); u'(R1) = 0 = u'{R2); where φ:= φ′: (-a;a) → R is an increasing homeomorphism with φ(0) = 0 and the continuous function φ: [-a; a] → R is of class C1 on (-a; a). The associated functional in the space of continuous functions over [R1; R2] 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.