Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents

Abstract

<p style='text-indent:20px;'>We consider the following nonlocal critical equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE145"> \begin{document}$\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\mu_1,\mu_2&lt;N $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M2">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M3">\begin{document}$ 4 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M4">\begin{document}$ N-4\leq\mu_1,\mu_2&lt;N $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M5">\begin{document}$ N\geq5 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 2_{\mu_{i}}^\ast: = \frac{N+\mu_i}{N-2}(i = 1,2) $\end{document}</tex-math></inline-formula> is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and <inline-formula><tex-math id="M7">\begin{document}$ I_{\mu_i} $\end{document}</tex-math></inline-formula> is the Riesz potential</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} I_{\mu_i}(x) = \frac{\Gamma(\frac{N-\mu_i}{2})}{\Gamma(\frac{\mu_i}{2})\pi^{\frac{N}{2}}2^{\mu_i}|x|^{N-\mu_i}}, \; i = 1,2, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with <inline-formula><tex-math id="M8">\begin{document}$ \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ s&gt;0 $\end{document}</tex-math></inline-formula>. Firstly, we prove the existence of the solutions of the equation (1). We also establish integrability and <inline-formula><tex-math id="M10">\begin{document}$ C^\infty $\end{document}</tex-math></inline-formula>-regularity of solutions and obtain the explicit forms of positive solutions via the method of moving spheres in integral forms. Finally, we show that the nondegeneracy of the linearized equation of (1) at <inline-formula><tex-math id="M11">\begin{document}$ U_0,V_0 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M12">\begin{document}$ \max\{\mu_1,\mu_2\}\rightarrow0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \min\{\mu_1,\mu_2\}\rightarrow N $\end{document}</tex-math></inline-formula>, respectively.</p>