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

Abstract

In this paper, we are concerned with the following elliptic equation $$(S{C}_{\epsilon })\{\begin{array}{ll}-\Delta u=|u{|}^{4/(n-2)}u[\ln (e+|u|){]}^{\epsilon }& in\; \Omega ,\\ u=0& on\partial \Omega ,\end{array}$$ where $\Omega$ 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.