Existence and Regularity of Minimizers of Nonconvex Functionals Depending onuand ∇u

We consider variational problems of the formmin∫Ω[f(Δu(x))+g(x,u(x))]dx:u∈u 0+H10(Ω),wheref: RN→[0,∞] is a possibly nonconvex function with quadratic growth at infinity andg(x,u) is Lipschitz continuous and strictly increasing (decreasing) inu. We prove the existence and local Lipschitz regularity of solutions for every boundary datumu0∈H1(Ω)∩L∞(Ω) on the basis of the structure of the epigraph of the convex envelope off. © 1998 Academic Press.