A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms

Abstract

In this article, we study a Hardy-Sobolev critical elliptic system involving coupled perturbation terms: (0.1) − Δ u + V 1 ( x ) u = η 1 η 1 + η 2 ∣ u ∣ η 1 − 2 u ∣ v ∣ η 2 ∣ x ′ ∣ + α α + β Q ( x ) ∣ u ∣ α − 2 u ∣ v ∣ β , − Δ v + V 2 ( x ) v = η 2 η 1 + η 2 ∣ v ∣ η 2 − 2 v ∣ u ∣ η 1 ∣ x ′ ∣ + β α + β Q ( x ) ∣ v ∣ β − 2 v ∣ u ∣ α , \left\{\begin{array}{c}-\Delta u+{V}_{1}\left(x)u=\frac{{\eta }_{1}}{{\eta }_{1}+{\eta }_{2}}\frac{{| u| }^{{\eta }_{1}-2}u{| v| }^{{\eta }_{2}}}{| x^{\prime} | }+\frac{\alpha }{\alpha +\beta }Q\left(x)| u{| }^{\alpha -2}u| v{| }^{\beta },\\ -\Delta v+{V}_{2}\left(x)v=\frac{{\eta }_{2}}{{\eta }_{1}+{\eta }_{2}}\frac{{| v| }^{{\eta }_{2}-2}v{| u| }^{{\eta }_{1}}}{| x^{\prime} | }+\frac{\beta }{\alpha +\beta }Q\left(x){| v| }^{\beta -2}v{| u| }^{\alpha },\end{array}\right. where n ≥ 3 n\ge 3 , 2 ≤ m < n 2\le m\lt n , x ≔ ( x ′ , x ″ ) ∈ R m × R n − m x:= \left(x^{\prime} ,{x}^{^{\prime\prime} })\in {{\mathbb{R}}}^{m}\times {{\mathbb{R}}}^{n-m} , η 1 , η 2 > 1 {\eta }_{1},{\eta }_{2}\gt 1 , and η 1 + η 2 = 2 ( n − 1 ) n − 2 {\eta }_{1}+{\eta }_{2}=\frac{2\left(n-1)}{n-2} , α , β > 1 \alpha ,\beta \gt 1 and α + β < 2 n n − 2 \alpha +\beta \lt \frac{2n}{n-2} , and V 1 ( x ) , V 2 ( x ) , Q ( x ) ∈ C ( R n ) {V}_{1}\left(x),{V}_{2}\left(x),Q\left(x)\in C\left({{\mathbb{R}}}^{n}) . Observing that (0.1) is doubly coupled, we first develop two efficient tools (i.e., a refined Sobolev inequality and a variant of the “Vanishing” lemma). On the previous tools, we will establish a global compactness result (i.e., a complete description for the Palais-Smale sequences of the corresponding energy functional) and some existence result for (0.1) via variational method. Our strategy turns out to be very concise because we avoid the use of Levy concentration functions and truncation techniques.