Existence of positive solutions for semilinear elliptic systems with indefinite weight

Abstract

This article concerns the existence of positive solutions of semilinear elliptic system $$displaylines{ -Delta u=lambda a(x)f(v),quadhbox{in }Omega,cr -Delta v=lambda b(x)g(u),quadhbox{in }Omega,cr u=0=v,quad hbox{on } partialOmega, }$$ where $Omegasubseteqmathbb{R}^N (Ngeq1)$ is a bounded domain with a smooth boundary $partialOmega$ and $lambda$ is a positive parameter. $a, b:Omegaomathbb{R}$ are sign-changing functions. $f, g:[0,infty)omathbb{R}$ are continuous with $f(0)>0$, $g(0)>0$. By applying Leray-Schauder fixed point theorem, we establish the existence of positive solutions for $lambda$ sufficiently small.

Keywords

• Semilinear elliptic systems
• indefinite weight
• positive solutions
• existence of solutions