Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space

Abstract

We study the Dirichlet problem with mean curvature operator in Minkowski spacediv(∇v1−|∇v|2)+λ[μ(|x|)vq]=0in B(R),v=0on ∂B(R), where λ>0 is a parameter, q>1, R>0, μ:[0,∞)→R is continuous, strictly positive on (0,∞) and B(R)={x∈RN:|x|<R}. Using upper and lower solutions and Leray–Schauder degree type arguments, we prove that there exists Λ>0 such that the problem has zero, at least one or at least two positive radial solutions according to λ∈(0,Λ), λ=Λ or λ>Λ. Moreover, Λ is strictly decreasing with respect to R.