Positive radial solutions to singular nonlinear elliptic problems involving nonhomogeneous operators

Abstract

We consider singular nonlinear elliptic problems involving nonhomogeneous operators on annular domains −div(A(|x|)B(|∇u|)∇u)=λK(|x|)f(u),x∈Ω,u=0,|x|=r1,a∂u∂n+c(λ,u)=0,|x|=r2,where λ>0, a≥0, N>1, Ω≔{x∈RN∣0<r1<|x|<r2<∞} and ∂u∂n is the outward normal derivative of u on ∂Br2. Here A∈C([r1,r2],(0,∞)), K∈C((r1,r2),(0,∞)), c∈C((0,∞)×R,R), B∈C([0,∞),[0,∞)) is such that B(s)s is a homeomorphism from [0,∞) onto [0,∞), and f∈C((0,∞),(0,∞)) has a singularity at 0. The aim of this paper is to analyze the existence and multiplicity of positive radial solutions according to the behavior of f near ∞. In particular, we discuss sufficient conditions for at least three positive radial solutions to exist. The results are obtained via a Krasnoselskii type fixed point theorem. Finally, we provide examples including Gelfand-type problems to illustrate each result.