On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled by (GRAPHICS) < Au, v - u > >= integral(Omega) vertical bar u(x)vertical bar(s-1) u(x)(v(x) - u(x))dx for any v is an element of H-0(1)(Omega), v <= psi u is an element of H-0(1)(Omega), u <= psi, where Omega is a bounded domain of R-N, N >= 3, with smooth boundary, A is the elliptic operator, riot in divergence form, given by Au = - Sigma(N)(i,j=1) D-i (a(ij)(x)D-j u) + Sigma(N)(i=1) a(i)(x)D(i)u + a(0)(x)u. Here a(ij), a(i), i, j = 1,...,N, and a(0) satisfy suitable regularity conditions, while 1 < s < 4/(N - 2) and the obstacle psi is a function sufficiently smooth. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a C-1,C-alpha-regularity result for the solutions of our problem.