Existence of solutions for quasilinear elliptic equations with nonlinear boundary conditions and indefinite weight

In this article, we establish the existence and non-existence of solutions for quasilinear equations with nonlinear boundary conditions and indefinite weight. Our proofs are based on variational methods and their geometrical features. In addition, we prove that all the weak solutions are in $C^{1,eta}(overline{Omega})$ for some $etain(0,1)$.