Semiclassical states for critical Choquard equations with critical frequency

Fashun Gao, Jiazheng Zhou

DOI: http://dx.doi.org/10.12775/TMNA.2020.001

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.

Keywords


Critical Choquard equation; Hardy-Littlewood-Sobolev inequality; semiclassical states; critical frequency

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