Symmetry and Regularity of Solutions to the Weighted Hardy–Sobolev Type System

Abstract

Abstract Hardy–Littlewood–Sobolev inequalities and the Hardy–Sobolev type system play an important role in analysis and PDEs. In this paper, we consider the very general weighted Hardy–Sobolev type system u ⁢ ( x ) = ∫ ℝ n 1 | x | τ ⁢ | x - y | n - α ⁢ | y | t ⁢ f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , v ⁢ ( x ) = ∫ ℝ n 1 | x | t ⁢ | x - y | n - α ⁢ | y | τ ⁢ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) ⁢ 𝑑 y , $u(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{\tau}|x-y|^{n-\alpha}|y|^{t}}f_{1}(u(y% ),v(y))dy,\quad v(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{t}|x-y|^{n-\alpha}|y|^% {\tau}}f_{2}(u(y),v(y))dy,$ where f 1 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 1 ⁢ u p 1 ⁢ ( y ) + μ 1 ⁢ v q 1 ⁢ ( y ) + γ 1 ⁢ u α 1 ⁢ ( y ) ⁢ v β 1 ⁢ ( y ) , $\displaystyle f_{1}(u(y),v(y))=\lambda_{1}u^{p_{1}}(y)+\mu_{1}v^{q_{1}}(y)+% \gamma_{1}u^{\alpha_{1}}(y)v^{\beta_{1}}(y),$ f 2 ⁢ ( u ⁢ ( y ) , v ⁢ ( y ) ) = λ 2 ⁢ u p 2 ⁢ ( y ) + μ 2 ⁢ v q 2 ⁢ ( y ) + γ 2 ⁢ u α 2 ⁢ ( y ) ⁢ v β 2 ⁢ ( y ) . $\displaystyle f_{2}(u(y),v(y))=\lambda_{2}u^{p_{2}}(y)+\mu_{2}v^{q_{2}}(y)+% \gamma_{2}u^{\alpha_{2}}(y)v^{\beta_{2}}(y).$ Only the special cases when γ 1 = γ 2 = 0 ${\gamma_{1}=\gamma_{2}=0}$ and one of λ i ${\lambda_{i}}$ and μ i ${\mu_{i}}$ is zero (for both i = 1 ${i=1}$ and i = 2 ${i=2}$ ) have been considered in the literature. We establish the integrability of the solutions to the above Hardy–Sobolev type system and the C ∞ ${C^{\infty}}$ regularity of solutions to this system away from the origin, which improves significantly the Lipschitz continuity in most works in the literature. Moreover, we also use the moving plane method of [8] in integral forms developed in [6] to prove that each pair ( u , v ) ${(u,v)}$ of positive solutions of the above integral system is radially symmetric and strictly decreasing about the origin.