具有上临界指数的p-Choquard方程基态解的存在性

Existence of Ground State Solution for p-Choquard Equation with Upper Critical Exponent

摘要: 该文讨论下列具有上临界指标的p-Choquard方程 $\{\begin{array}{l}-{\Delta }_{p}u+V\left(x\right){\left|u\right|}^{p-2}u=\left[{\left(-{\Delta }^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{p_{\alpha }^{\ast }}\right]{\left|u\right|}^{p_{\alpha }^{\ast }-2}u,in{ℝ}^N;\\ u\in W^{1,p}\left({ℝ}^N\right),\end{array}$ 其中$N\ge 3$ ，$p_{\alpha }^{*}=\frac{p}{2}\left(\frac{N+\alpha }{N-p}\right)$ 是关于Hardy-Littlewood-Sobolev不等式的上临界指数，$2\le p<N$ ，${\Delta }_p:=div\left({\left|\nabla u\right|}^{p-2}\nabla u\right)$ ，${\left(-{\Delta }^D\right)}^{\frac{-\alpha }{2}}$ 是关于Dirichlet边界条件下的Riesz变分位势，指数$\alpha \in \left(0,N\right)$ 。该文通过Nehari流形和山路引理证明上述方程存在基态解。

Abstract: In this paper, we are concerned with the following p-Choquard equation with upper critical exponent: $\{\begin{array}{l}-{\Delta }_{p}u+V\left(x\right){\left|u\right|}^{p-2}u=\left[{\left(-{\Delta }^D\right)}^{\frac{-\alpha }{2}}\ast {\left|u\right|}^{p_{\alpha }^{\ast }}\right]{\left|u\right|}^{p_{\alpha }^{\ast }-2}u,in{ℝ}^N;\\ u\in W^{1,p}\left({ℝ}^N\right),\end{array}$ where $N\ge 3$ , $p_{\alpha }^{*}=\frac{p}{2}\left(\frac{N+\alpha }{N-p}\right)$ is the upper critical Hardy-Littlewood-Sobolev exponent, $2\le p<N$ , ${\Delta }_p:=div\left({\left|\nabla u\right|}^{p-2}\nabla u\right)$ , ${\left(-{\Delta }^D\right)}^{\frac{-\alpha }{2}}$ is the Riesz fractional potential with Dirichlet boundary condition. $\alpha \in \left(0,N\right)$ . We study the existence of the ground state solution to above equation by using the methods of Nehari manifold and the Mountain Pass Lemma.