Existence of Solutions of a Superlinear Elliptic System at Resonance

Abstract

We are concerned with the existence of solution for elliptic system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u+\delta v+f(u^{+})+k_{1}(x)&{} \quad \textrm{in}\ \Omega ,\\ -\Delta v=\delta u+\gamma v+g(v^{+})+k_{2}(x)&{}\quad \textrm{in}\ \Omega ,\\ u=v=0&{}\quad \textrm{on}\ \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^{N}$, $N\ge 3$, is a smooth bounded domain, $s^{+}:=\max \{s,0\}$, $\lambda ,\delta$ and $\gamma$ are real parameters such that $\max \{\lambda ,\gamma \}>0$ and $\delta >0$. By topological degree arguments, we show the existence of nontrivial solutions for above system under certain superlinear growth conditions on f, g and an one-sided Landesman−Lazer condition on $k_{1},k_{2}$. Also, a priori bound for the solutions are obtained by adapting the method of Brezis−Turner.