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

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 uu 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 uu is differentiable and its gradient is αα-Hőlder continuous on Ω¯ for any α∈(0,1)α∈(0,1).