Multiple solutions to a singular Lane-Emden-Fowler equation with convection term

Abstract

This article concerns the existence of multiple solutions for the problem −∆u = K(x)u −α + s(Au β + B|∇u| ζ ) + f(x) in Ω u > 0 in Ω u = 0 on ∂Ω , where Ω is a smooth, bounded domain in Rn with n ≥ 2, α, β, ζ, A, B and s are real positive numbers, and f(x) is a positive real valued and measurable function. We start with the case s = 0 and f = 0 by studying the structure of the range of −u α∆u. Our method to build K’s which give at least two solutions is based on positive and negative principal eigenvalues with weight. For s small positive and for values of the parameters in finite intervals, we find multiplicity via estimates on the bifurcation set.