Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

Abstract

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy–Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α -Hőlder continuous on Ω ¯ for any α ∈ ( 0 , 1 ) .