Abstract

We study the existence, nonexistence and multiplicity of non-negative solutions for the family of problems where is a bounded domain in 2 and > 0 is a parameter. The coefficient a(x) is permitted to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the TrudingerMoser inequality and variational methods. Note that when f(x, u) = 0 the equation is of Liouville type. © 2009 Copyright Royal Society of Edinburgh.