Ground States for a Nonlinear Elliptic Equation Involving Multiple Hardy–Sobolev Critical Exponents

Abstract

Abstract The present paper studies the nonlinear elliptic equation { Δ ⁢ u + λ | x | s ⁢ u p - 1 + ∑ i = 1 l λ i | x | s i ⁢ u 2 ∗ ⁢ ( s i ) - 1 + u 2 ∗ - 1 = 0 in ⁢ Ω , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} \displaystyle\Delta u+\frac{\lambda}{|x|^{s}}u^{p-1}+% \sum_{i=1}^{l}\frac{\lambda_{i}}{|x|^{s_{i}}}u^{2^{\ast}(s_{i})-1}+u^{2^{\ast}% -1}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.$ which involves multiple Hardy–Sobolev critical exponents, where λ ≠ 0 ${\lambda\neq 0}$ , s ∈ [ 0 , 2 ) ${s\in[0,2)}$ , p ∈ ( 2 , 2 ∗ ⁢ ( s ) ) ${p\in(2,2^{\ast}(s))}$ , 0 < s 1 < ⋯ < s l < 2 $0<s_{1}<\cdots<s_{l}<2$ , λ 1 , … , λ k > 0 ${\lambda_{1},\dots,\lambda_{k}>0}$ , λ k + 1 , … , λ l < 0 ${\lambda_{k+1},\dots,\lambda_{l}<0}$ for some k ∈ [ 1 , l ] ${k\in[1,l]}$ and Ω is a C 1 $C^{1}$ open bounded domain in ℝ N $\mathbb{R}^{N}$ , N ≥ 3 ${N\geq 3}$ , containing the origin. The existence of a positive ground state solution is established when λ > 0 ${\lambda>0}$ and p ≥ 2 ∗ ⁢ ( s k ) ${p\geq 2^{\ast}(s_{k})}$ .