Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).

Multiple positive solutions for a nonlinear elliptic equation involving Hardy-Sobolev-Maz’ya term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz'ya term:
-Δu-λ u/|y|² = (|u|^pt-1u)/|y|^t + μf(x), x∈ Ω,
where Ω is a bounded domain in R^N(N ≥ 2), 0 ∈ Ω, x = (y, z) ∈ R^k ×R^N-k and p_t = (N+2-2t)/(N-2) (0 ≤ t ≤ 2). For f(x) ∈ C¹(Ω)\{0}, we show that there exists a constant μ^* >0 such that the problem possesses at least two positive solutions if μ ∈ (0, μ^*) and at least one positive solution if μ = μ^*. Furthermore, there are no positive solutions if μ ∈ (μ^*,+∞).