Regularity of the solutions to a nonlinear boundary problem with indefinite weight

In this paper we study the regularity of the solutions to the problemDelta_p u = |u|^{p 2}u in the bounded smooth domainOmega R^N,with| u|^{p 2} partial_{nu} u = lambda V (x)|u|^{p 2}u + h as a nonlinear boundary condition, where partial Omega is C^{2,beta}, with beta ∈]0, 1[, and V is a weight in L^s(partial Omega) and h ∈ L^s(partial Omega ) for some s ≥ 1. We prove that all solutions are in L^{infty}(Omega) cap L^{infty}(Omega), and using the D.Debenedetto’s theorem of regularity in [1] we conclude that those solutions are in C^{1,alpha} overline{Omega}) for some alpha ∈ ]0, 1[.