Normalized Solutions to Schrödinger Equations with Critical Exponent and Mixed Nonlocal Nonlinearities

We study the existence and nonexistence of normalized solutions \((u_a, \lambda _a)\in H^{1}(\mathbb {R}^N)\times \mathbb {R}\) $(u_a,{\lambda }_a)\in H^1({\mathbb{R}}^N)\times \mathbb{R}$ to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities: \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda u+ (I_{\alpha } * \vert u \vert ^p) \vert u \vert ^{p-2}u+\mu (I_{\alpha } * \vert u \vert ^q) \vert u \vert ^{q-2}u\quad \text {in }\mathbb {R}^N,\\ \int _{\mathbb {R}^{N}} \vert u \vert ^2 \textrm{d} x=a^2>0, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u=\lambda u+(I_{\alpha }{\ast |u|}^p{)|u|}^{p-2}u+\mu (I_{\alpha }{\ast |u|}^q{)|u|}^{q-2}u\text{in}{\mathbb{R}}^N,\hfill \\ {\int }_{{\mathbb{R}}^N}{|u|}^2\text{d}x=a^2>0,\hfill \end{array}\right)\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(\alpha \in (0,N)\) $\alpha \in (0,N)$ , \(\mu \in \mathbb {R}\) $\mu \in \mathbb{R}$ , \(\frac{N+\alpha }{N}< q< p \le \frac{N+\alpha }{N-2}\) $\frac{N+\alpha }{N}<q<p\le \frac{N+\alpha }{N-2}$ , and \(I_{\alpha }\) $I_{\alpha }$ is the Riesz potential. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to nonlocal Schrödiger equations with a fixed \(L^2\) $L^2$ -norm \(\Vert u\Vert _2=a>0\) ${\Vert u\Vert }_2=a>0$ . The leading term is \(L^2\) $L^2$ -supercritical, that is, \(p\in (\frac{N+\alpha +2}{N},\frac{N+\alpha }{N-2}]\) $p\in (\frac{N+\alpha +2}{N},\frac{N+\alpha }{N-2}]$ , where the Hardy–Littlewood–Sobolev critical exponent \(p=\frac{N+\alpha }{N-2}\) $p=\frac{N+\alpha }{N-2}$ appears. We first prove that there exist two normalized solutions if \(q\in (\frac{N+\alpha }{N},\frac{N+\alpha +2}{N})\) $q\in (\frac{N+\alpha }{N},\frac{N+\alpha +2}{N})$ with \(\mu >0\) $\mu >0$ small, that is, one is at the negative energy level while the other one is at the positive energy level. For \(q=\frac{N+\alpha +2}{N}\) $q=\frac{N+\alpha +2}{N}$ , we show that there is a normalized ground state for \(0<\mu <\tilde{\mu } \) $0<\mu <\tilde{\mu }$ and there exist no ground states for \(\mu >\tilde{\mu }\) $\mu >\tilde{\mu }$ , where \(\tilde{\mu }\) $\tilde{\mu }$ is a sharp positive constant. If \(q\in (\frac{N+\alpha +2}{N},\frac{N+\alpha }{N-2})\) $q\in (\frac{N+\alpha +2}{N},\frac{N+\alpha }{N-2})$ , we deduce that there exists a normalized ground state for any \(\mu >0\) $\mu >0$ . We also obtain some existence and nonexistence results for the case \(\mu <0\) $\mu <0$ and \(q\in (\frac{N+\alpha }{N},\frac{N+\alpha +2}{N}]\) $q\in (\frac{N+\alpha }{N},\frac{N+\alpha +2}{N}]$ . Besides, we analyze the asymptotic behavior of normalized ground states as \(\mu \rightarrow 0^{+}\) $\mu \to 0^{+}$ .