Classification, non-degeneracy and existence of solutions to nonlinear Choquard equations

In this paper, we first study the classification problem of positive solutions to the nonlocal Choquard equation \(\begin{aligned} (-\Delta )^{\alpha }u=\left( \textrm{I}_{\beta }*\left| u\right| ^{p}\right) \left| u\right| ^{p-2}u, \qquad u\in {\mathcal {D}}^{\alpha ,2}\left( {\mathbb {R}}^N\right) \cap L^{\frac{2Np}{N+\beta }}\left( {\mathbb {R}}^N\right) , \end{aligned}\) $$\begin{array}{c}\hfill {(-\mathrm{\Delta })}^{\alpha }u=\left({\text{I}}_{\beta },,\ast ,{\left|u\right|}^p\right){\left|u\right|}^{p-2}u,u\in {\mathcal{D}}^{\alpha ,2}\left({\mathbb{R}}^N\right)\cap L^{\frac{2Np}{N+\beta }}\left({\mathbb{R}}^N\right),\end{array}$$ where \(\alpha \in \left( 0,\frac{N}{2}\right) \) $\alpha \in \left(0,,,\frac{N}{2}\right)$ with \(N\ge 1\) $N\ge 1$ , \(\textrm{I}_{\beta }(x)\) ${\text{I}}_{\beta }(x)$ is the standard Riesz potential with \(\beta \in \left( 0,N\right) \) $\beta \in \left(0,,,N\right)$ if \(N\le 4\alpha \) $N\le 4\alpha$ , or \(\beta \in \left[ N-4\alpha ,N\right) \) $\beta \in \left[N,-,4,\alpha ,,,N\right)$ if \(N>4\alpha \) $N>4\alpha$ , and \(p\in [2, 2^*_{\alpha ,\beta }]\) $p\in [2,2_{\alpha ,\beta }^{\ast }]$ with \(2^*_{\alpha ,\beta }:=\frac{N+\beta }{N-2\alpha }\) $2_{\alpha ,\beta }^{\ast }:=\frac{N+\beta }{N-2\alpha }$ which denotes the corresponding Hardy–Littlewood–Sobolev critical exponent. By applying the method of moving planes, we prove that the above Choquard equation has no positive solution in the subcritical case \(p\in [2, 2^*_{\alpha ,\beta })\) $p\in [2,2_{\alpha ,\beta }^{\ast })$ , and any positive solution must be of the form \(\begin{aligned} U_{\lambda , x_0}(x)=C_{\alpha ,\beta ,N}\left( \frac{\lambda }{\lambda ^2 +\left| x-x^0\right| ^2}\right) ^{\frac{N-2\alpha }{2}} \end{aligned}\) $$\begin{array}{c}\hfill U_{\lambda ,x_0}(x)=C_{\alpha ,\beta ,N}{\left(\frac{\lambda }{{\lambda }^2+{\left|x,-,x^0\right|}^2}\right)}^{\frac{N-2\alpha }{2}}\end{array}$$ in the critical case \(p=2^*_{\alpha ,\beta }\) $p=2_{\alpha ,\beta }^{\ast }$ , where \(x^0\in {\mathbb {R}}^N\) $x^0\in {\mathbb{R}}^N$ , \(\lambda >0\) $\lambda >0$ and the constant \(C_{\alpha ,\beta ,N}>0\) $C_{\alpha ,\beta ,N}>0$ only depends on \(\alpha \) $\alpha$ , \(\beta \) $\beta$ and N. Moreover, we prove that this solution \(U_{\lambda , x^0}(x)\) $U_{\lambda ,x^0}(x)$ is non-degenerate. Based on the non-degeneracy result, we study the existence of solutions to the following Choquard equation with potential: \(\begin{aligned} -\Delta u+V(\left| x\right| )u=\left( \textrm{I}_{\beta }*u^{2_{1, \beta }^*}\right) u^{2_{1,\beta }^*-1},\qquad u\in {\mathcal {D}}^{1,2}({\mathbb {R}}^N), \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+V(\left|x\right|)u=\left({\text{I}}_{\beta },,\ast ,u^{2_{1,\beta }^{\ast }}\right)u^{2_{1,\beta }^{\ast }-1},u\in {\mathcal{D}}^{1,2}({\mathbb{R}}^N),\end{array}$$ where \(\beta \in [N-4,N-2)\) $\beta \in [N-4,N-2)$ , \(N\ge 5\) $N\ge 5$ , and \(V(\cdot )\) $V(·)$ is a bounded non-negative function. We show that this Choquard equation possesses infinitely many non-radial solutions provided that \(r^{2}V(r)\) $r^{2}V(r)$ has an isolated local maximum point, or isolated local minimum point \(r_0>0\) $r_0>0$ satisfying \(V(r_0)>0\) $V(r_0)>0$ .