Variational Inequalities with the Logistic Type Nonlinearities and Dependence on the Gradient

In this paper, we study the following variational inequality {(u is an element of K,)(< Au,v-u >) + integral(Omega) g(x,u)(v-u)>= integral(Omega) f(x,u,del u)(v-u),for all v is an element of K, where K = {u. W-0(1,p) (Omega) : u(x) >= 0}, A is the p- Laplacian and the function g is increasing in the second variable. By constructing the solution operator for an associate variational inequality, we reduce the problem to a fixed point equation. Then, we apply the fixed point index to prove the existence of the nontrivial solution of the problem.