A Note on the Sign-Changing Solutions for a Double Critical Hardy–Sobolev–Maz’ya Problem

Abstract

Abstract In this paper, we investigate the following double critical Hardy–Sobolev–Maz’ya problem: { - Δ ⁢ u = μ ⁢ | u | 2 * ⁢ ( t ) - 2 ⁢ u | y | t + | u | 2 * ⁢ ( s ) - 2 ⁢ u | y | s + a ⁢ ( x ) ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} &\displaystyle{-}\Delta u=\mu\frac{|u|^{2^{*}(t)-2}u}{|% y|^{t}}+\frac{|u|^{2^{*}(s)-2}u}{|y|^{s}}+a(x)u&&\displaystyle\text{in }\Omega% ,\\ &\displaystyle u=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right.$ where μ ≥ 0 ${\mu\geq 0}$ , a ⁢ ( x ) > 0 ${a(x)>0}$ , 2 * ⁢ ( t ) = 2 ⁢ ( N - t ) N - 2 ${2^{*}(t)=\frac{2(N-t)}{N-2}}$ , 2 * ⁢ ( s ) = 2 ⁢ ( N - s ) N - 2 ${2^{*}(s)=\frac{2(N-s)}{N-2}}$ , 0 ≤ t < s < 2 ${0\leq t<s<2}$ , x = ( y , z ) ∈ ℝ k × ℝ N - k ${x=(y,z)\in\mathbb{R}^{k}\times\mathbb{R}^{N-k}}$ , 2 ≤ k < N , ( 0 , z * ) ∈ Ω ¯ $2\leq k<N,(0,z^{*})\in\bar{\Omega}$ and Ω is an open bounded domain in ℝ N ${\mathbb{R}^{N}}$ . By applying an abstract theorem presented in [42], we prove that if N > 6 + t ${N>6+t}$ when μ > 0 , ${\mu>0,}$ and N > 6 + s ${N>6+s}$ when μ = 0 , ${\mu=0,}$ and Ω satisfies some geometric conditions, then the above problem has infinitely many sign-changing solutions. The main tool is to estimate the Morse indices of these nodal solutions.