带对数项的非局部Choquard方程解的存在性

Existence of Solutions to Non-Local Choquard Equations with Logarithmic Terms

摘要: 我们主要关注如下非局部Choquard方程解的存在性：$-\Delta u=\left({\displaystyle {\int }_{\Omega }\frac{{\left|u\right|}^{2_{\mu }^{\ast }}}{{\left|x-y\right|}^{\mu }}\text{d}y}\right){\left|u\right|}^{2_{\mu }^{\ast }-2}u+\lambda \left({\displaystyle {\int }_{\Omega }\frac{{\left|u\right|}^q}{{\left|x-y\right|}^{\mu }}\text{d}y}\right){\left|u\right|}^{q-2}u+\beta u\log u^2\text{in}\Omega$ 这里$\Omega$ 是${ℝ}^N$ 中一个具有光滑边界的有界区域，$\lambda ,\beta >0$ 为实参数，$2<q<2_{\mu }^{\ast }$ ，$2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}\left(N\ge 5\right)$ 是Hardy-Littlewood-Sobolev不等式意义下的上临界指标。

Abstract: We are interested in the existence of the following nonlocal Choquard equation: $-\Delta u=\left({\displaystyle {\int }_{\Omega }\frac{{\left|u\right|}^{2_{\mu }^{\ast }}}{{\left|x-y\right|}^{\mu }}\text{d}y}\right){\left|u\right|}^{2_{\mu }^{\ast }-2}u+\lambda \left({\displaystyle {\int }_{\Omega }\frac{{\left|u\right|}^q}{{\left|x-y\right|}^{\mu }}\text{d}y}\right){\left|u\right|}^{q-2}u+\beta u\log u^2\text{in}\Omega$ where $\Omega$ is a bounded domain of ${ℝ}^N$ with smooth boundary, $\lambda ,\beta >0$ are real parameters, $2<q<2_{\mu }^{\ast }$ , $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}\left(N\ge 5\right)$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.