Title: Non existence of solutions for a slightly super-critical elliptic problem with non-power nonlinearity Show Chinese title

Title: 略超临界椭圆问题非幂次非线性解的不存在性

Abstract: In this paper, we are concerned with the following elliptic equation $$ ( SC_\varepsilon ) \qquad \begin{cases} -\Delta u = |u|^{4/(n-2)}u [\ln (e+|u|)]^\varepsilon & \hbox{ in } \Omega,\\ u = 0 & \hbox{ on }\partial \Omega, \end{cases} $$ where $\Omega $ is a smooth bounded open domain in $\mathbb{R}^n, \ n\geq 3$ and $\varepsilon >0$. In Comm. Contemp. Math. (2003), Ben Ayed et al. showed that the slightly supercritical usual elliptic problem has no single peaked solution. Here we extend their result for problem $( SC_\varepsilon )$ when $\varepsilon$ is small enough, and that by assuming a new assumption. Show Chinese abstract

Abstract: 在本文中，我们关注以下椭圆方程$$ ( SC_\varepsilon ) \qquad \begin{cases} -\Delta u = |u|^{4/(n-2)}u [\ln (e+|u|)]^\varepsilon & \hbox{ in } \Omega,\\ u = 0 & \hbox{ on }\partial \Omega, \end{cases} $$，其中$\Omega $是$\mathbb{R}^n, \ n\geq 3$中的一个光滑有界开区域，且$\varepsilon >0$。 在 Comm. Contemp. Math. (2003) 中，Ben Ayed 等人表明，略微超临界的通常椭圆问题没有单峰解。 在这里，我们当$\varepsilon$足够小时，将他们的结果扩展到问题$( SC_\varepsilon )$，并且通过假设一个新假设。