Multiple Solutions for an Elliptic Equation with Hardy-Sobolev Critical Exponent, Hardy-Sobolev-Maz’ya Potential and Sign-Changing Weights

In the present paper, an elliptic equation with Hardy-Sobolev critical exponent, Hardy-Sobolev-Maz’ya potential and sign-changing weights, is considered. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained.

带有Hardy-Sobolev-Maz'ya项及Hardy-Sobolev临界指数的半线性椭圆方程正解的存在性

研究奇异的半线性椭圆方程-Δu-μ(u/|y|~2)=|u|^(2*(s)-2)u/|y|~s+λu运用山路引理,得到了其正解的存在性.

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 RN（N ≥ 2）, 0 ∈ Ω, x = （y, z） ∈ Rk ×RN-k and 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 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＊,＋∞）.