On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents

By Fashun Gao, Minbo Yang. 2016

155 citations recorded by the source corpus.

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, $$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu<N$ and $2_{\mu}^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5adeb864b6bffe6fc7416fb488c5ab2da87012f2

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent (2017, background): “The upper critical exponent plays a similar role as the Sobolev critical exponent in the local semilinear equations [4,9].”
Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth (2018, unknown): “This turns out to be a Lions-type problem and it is considered in a companion paper [26] as it involves quite different techniques.”
Standing waves to upper critical Choquard equation with a local perturbation: Multiplicity, qualitative properties and stability (2021, background): “Moreover, similar to that [39] and [24], by direct computation, we have I u u A N C N S O ε .”