Multiplicity and uniqueness for Lane-Emden equations and systems with Hardy potential and measure data

Abstract

Let Ω be a C2 bounded domain in RN (N≥3), δ(x)=dist(x,∂Ω) and CH(Ω) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form−Δu−μδ2u=up in Ω,u=ρν on ∂Ω,(Pρ) where 0<μ<CH(Ω), ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1<p<Nμ, with Nμ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ⁎ such that problem (Pρ) admits a positive solution if 0<ρ≤ρ⁎, and no positive solution if ρ>ρ⁎. In this paper, we go further in the study of the solution set of (Pρ). We show that the problem admits at least two positive solutions if 0<ρ<ρ⁎ and a unique positive solution if ρ=ρ⁎.We also prove the existence of at least two positive solutions for Lane-Emden systems{−Δu−μδ2u=vp in Ω,−Δv−μδ2v=uq in Ω,u=ρν,v=στ on ∂Ω, under the smallness condition on the positive parameters ρ and σ.