Abstract
We study the multiplicity of semiclassical states for the Choquard equation $$ -\varepsilon^2\Delta u +V(x)u =\varepsilon^{\mu-N}\bigg(\int_{\mathbb{R}^{N}} \frac{G(y,u(y))}{|x-y|^\mu}dy\bigg)g(x,u) \quad \mbox{in $\mathbb{R}^{N}$}, $$ where $0< \mu< N$, $N\geq3$, $\varepsilon$ is a positive parameter and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. The potential function $V$ is assumed to be nonnegative with $V(x)=0$ in some region of $\mathbb{R}^{N}$. Using the genus theory we prove the multiplicity of semiclassical states for the critical Choquard equation.
Citation
Fashun Gao. Jiazheng Zhou. "Semiclassical states for critical Choquard equations with critical frequency." Topol. Methods Nonlinear Anal. 57 (1) 107 - 133, 2021. https://doi.org/10.12775/TMNA.2020.001
Information