On the existence of multiple positive solutions to some superlinear systems

Abstract

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the form -Δ u=g 1(x,u,v), -Δv=g 2(x,u,v), on a smooth, bounded domain Ω⊂ℝ n with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave-convex shape to their nonlinearity. © 2012 The Royal Society of Edinburgh.