Non existence of solutions for a slightly super-critical elliptic problem with non-power nonlinearity

In this paper, we are concerned with the following elliptic equation $$(S{C}_{\epsilon })\{\begin{array}{ll}-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{4\mathrm{/}(n-2)}u[\ln (e+\mathrm{\mid }u\mathrm{\mid }){]}^{\epsilon }& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }$ is a smooth bounded open domain in ${\mathbb{R}}^n,n\ge 3$ and $\epsilon >0$. 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 $(S{C}_{\epsilon })$ when $\epsilon$ is small enough, and that by assuming a new assumption.