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 $$( SC_\varepsilon ) \qquad \begin{cases} -\Delta u = |u|^{4/(n-2)}u [\ln (e+|u|)]^\varepsilon & \hbox{ in } \Omega,\\ u = 0 & \hbox{ on }\partial \Omega, \end{cases}$$ 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.