Uniqueness and asymptotic behavior of positive solution of quasilinear elliptic equations with Hardy potential

Abstract

In this paper, we study the uniqueness and singular behavior at the origin of positive solution to −Δpu=λup−1|x|p−|x|σf(u),x∈Ω∖{0},where N>p>1, σ>−p, 0∈Ω and Ω⊂RN(N≥3) is a domain. For the nonlinear term f(u), we suppose that limu→∞f(u)uq=a>0 and limu→0+f(u)us=b. When Ω=RN, σ≥0 and σ>N(s−p+1)−spp−1, we discuss the uniqueness of positive solution. When Ω is a bounded smooth domain, σ≥0 and σ>N(q−p+1)−pqp−1, we prove that each positive solution u(x) satisfies lim|x|→0u(x)|x|p+σq−p+1=λa+(p+σq−p+1)p−1(1+p+σq−p+1)p−1a−N−1a1q−p+1,and the equation with the boundary condition u=ϕ≥0 on ∂Ω has a unique positive solution. When Ω=BR(0) and σ≥0, the uniqueness of positive solution to the corresponding Dirichlet problem is also established.