A saddle point type solution for a system of operator equations

Abstract

Let $\Omega \subset \mathbb{R}^{n}$, $n>1$ and let $p,q\geq 2$. We consider the system of nonlinear Dirichlet problems \begin{equation*} \left\lbrace \begin{aligned} (Au)(x)& =N_{u}^{\prime }(x,u(x),v(x)), &&x\in\Omega, \\ -(Bv)(x)& =N_{v}^{\prime }(x,u(x),v(x)), &&x\in\Omega, \\ u(x) &= 0, &&x\in \partial \Omega,\\ v(x) &= 0, && x\in \partial \Omega, \end{aligned}\right. \end{equation*} where $N:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ is $C^1$ and is partially convex-concave and $A:W^{1,p}_0 (\Omega) \rightarrow(W^{1,p}_0 (\Omega) )^*$, $B: W^{1,p}_0 (\Omega) \rightarrow(W^{1,p}_0 (\Omega) )^*$ are monotone and potential operators. The solvability of this system is reached via the Ky–Fan minimax theorem.

Keywords

• ky–fan minimax theorem
• dirichlet problem
• potential operators
• monotone operators