Multiple positive solutions for classes of elliptic systems with combined nonlinear effects

Abstract

We study the existence of multiple positive solutions to systems of the form \begin{equation*} \begin{cases} \qquad-{\Delta} u ={\lambda} f(v), & \text{ in }{\Omega},\\ \qquad-{\Delta} v ={\lambda} g(u), & \text{ in }{\Omega},\\ \qquad\quad~~ u=0=v, & \text{ on }{\partial}{\Omega}. \end{cases} \end{equation*} Here ${\Delta}$ is the Laplacian operator, ${\lambda}$ is a positive parameter, ${\Omega}$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary and $f, g$ belong to a class of positive functions that have a combined sublinear effect at $\infty$. Our results also easily extend to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.