Variational Inequalities With The Logistic Type Nonlinearities And Dependence On The Gradient

Abstract

In this paper, we study the following variational inequality
\begin{equation*}
\left\{
\begin{aligned}
&u \in K,\\
&\langle Au,v-u\rangle + \int \limits _{\Omega}g(x,u)(v-u) \geq \int \limits_{\Omega} f(x,u,\nabla u)(v-u), \forall v \in K,
\end{aligned}\right.
\end{equation*}
where $K=\{u \in W_0^{1,p}(\Omega): u(x)\ge 0\}$, $A$ 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.