Multiple normalized solutions for the fractional choquard equation with Hardy-Littlewood-Sobolev upper critical exponent
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This paper considers the existence of multiple normalized solutions to the following fractional choquard equation with Hardy-\linebreak Littlewood-Sobolev upper critical exponent \[ \left.\left\{\begin{array}{ll}(-\Delta)^s u+V(\varepsilon x)u=\lambda u+\left(I_\alpha*|u|^p\right)|u|^{p-2}u\\ \hspace{2.8cm}+\mu\left(I_\alpha*|u|^{2^*_{\alpha,s}}\right)|u|^{{2^*_{\alpha,s}}-2}u,&\ \text{in } \mathbb{R}^N,\\\int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a^2,\end{array}\right.\right. \] where $0<s<1$, $0<\alpha<N$, $\varepsilon, \mu, a> 0$, $N\geq2s$, $\lambda \in{\mathbb{R}}$, $\frac{N+\alpha}N<p<\frac{N+2s+\alpha}N<2_{\alpha,s}^*,$ $2_{\alpha,s}^*:=\frac{N+\alpha}{N-2s}$ is the upper Hardy-Littlewood-Sobolev critical exponent, $I_\alpha$ is the Riesz potential, $V:\mathbb{R}^{N}\to[0,\infty)$ is a continuous function. Since it has local minimizer, we will use the truncated skill to deal with critical term. When $\varepsilon$ is small enough, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V$ attains its minimum value by applying Lusternik-Schnirelmann category.
Abstract
This paper considers the existence of multiple normalized solutions to the following fractional choquard equation with Hardy-\linebreak Littlewood-Sobolev upper critical exponent \[ \left.\left\{\begin{array}{ll}(-\Delta)^s u+V(\varepsilon x)u=\lambda u+\left(I_\alpha*|u|^p\right)|u|^{p-2}u\\ \hspace{2.8cm}+\mu\left(I_\alpha*|u|^{2^*_{\alpha,s}}\right)|u|^{{2^*_{\alpha,s}}-2}u,&\ \text{in } \mathbb{R}^N,\\\int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a^2,\end{array}\right.\right. \] where $0<s<1$, $0<\alpha<N$, $\varepsilon, \mu, a> 0$, $N\geq2s$, $\lambda \in{\mathbb{R}}$, $\frac{N+\alpha}N<p<\frac{N+2s+\alpha}N<2_{\alpha,s}^*,$ $2_{\alpha,s}^*:=\frac{N+\alpha}{N-2s}$ is the upper Hardy-Littlewood-Sobolev critical exponent, $I_\alpha$ is the Riesz potential, $V:\mathbb{R}^{N}\to[0,\infty)$ is a continuous function. Since it has local minimizer, we will use the truncated skill to deal with critical term. When $\varepsilon$ is small enough, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V$ attains its minimum value by applying Lusternik-Schnirelmann category.