ABSTRACT

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem $\begin{array}{}\text{{}(P{)}_{\delta }\text{}}& \begin{array}{rl}& -\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right)={\left({\int }_{\mathrm{\Omega }}F(x,u)\right)}^{r}f(x,u)+\delta |u{|}^{p^{\ast }-2}u\text{in}\mathrm{\Omega },\\ & u\ge 0\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where Ω is a smooth bounded domain of ${\mathbb{R}}^N$, $N\ge 2$, $\delta =0$ or $\delta =1,p_i$ and $r\ge 0$, $1<p_1\le \cdots \le p_N<p^{\ast }$ are parameters where $p^{\ast }=N\overline{p}/(N-\overline{p})$ is a critical exponent, $\overline{p}=N/\sum _{i=1}^{N}1/p_i$ and $\overline{p}<N$. The nonlinearity $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ can be discontinuous, has subcritical growth subcritical growth and $F(x,t)={\int }_0^{t}f(x,s)\mathrm{d}s$. Under appropriate assumptions on f, we obtain the existence of nontrivial solutions for $(P{)}_{\delta }$ using the Ekeland's Variational Principle, Nonsmooth Mountain Pass Theorem and a Concentration Compactness-Principle.

摘要

在本文中，我们对各向异性非局部问题的解的存在性感兴趣 $\begin{array}{}\text{{}（P{）}_{\delta }\text{}}& \begin{array}{rl}& -\sum _{\mathrm{一世}=\mathrm{1个}}^{ñ}\frac{\mathrm{\partial }}{\mathrm{\partial }{X}_{\mathrm{一世}}}\left({\left|\frac{\mathrm{\partial }ü}{\mathrm{\partial }{X}_{\mathrm{一世}}}\right|}^{p_{\mathrm{一世}}-\mathrm{2个}}\frac{\mathrm{\partial }ü}{\mathrm{\partial }{X}_{\mathrm{一世}}}\right)={\left({\int }_{\mathrm{\Omega }}F（X，ü）\right)}^{\mathrm{[R}}F（X，ü）+\delta |ü{|}^{p^{\ast }-\mathrm{2个}}ü\text{在}\mathrm{\Omega }，\\ & ü\ge 0\text{在}\mathrm{\Omega }，ü=0\text{上}\mathrm{\partial }\mathrm{\Omega }，\end{array}\end{array}$ 其中Ω是 ${\mathbb{[R}}^{ñ}$， $ñ\ge \mathrm{2个}$， $\delta =0$ 或者 $\delta =\mathrm{1个}，p_{\mathrm{一世}}$ 和 $\mathrm{[R}\ge 0$， $\mathrm{1个}<p_{\mathrm{1个}}\le \cdots \le p_{ñ}<p^{\ast }$ 是参数，其中 $p^{\ast }=ñ\overline{p}/（ñ-\overline{p}）$ 是一个关键的指数， $\overline{p}=ñ/\sum _{\mathrm{一世}=\mathrm{1个}}^{ñ}\mathrm{1个}/p_{\mathrm{一世}}$ 和 $\overline{p}<ñ$。非线性$F：\mathrm{\Omega }\times \mathbb{[R}\to \mathbb{[R}$ 可以是不连续的，具有亚临界增长和亚临界增长， $F（X，Ť）={\int }_0^{Ť}F（X，s）\mathrm{d}s$。在f的适当假设下，我们得到存在非平凡解的存在$（P{）}_{\delta }$ 使用Ekeland的变分原理，不光滑的山口定理和浓度紧凑性原理。