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In this paper, we deal with the boundary value problem $-\Delta u=|u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon$ in a bounded smooth domain $\Omega$in $\mathbb{R}^n$, $n\geq 3$ with homogenous Dirichlet boundary condition. Here$\varepsilon>0$. Clapp et al. in Journal of Diff. Eq. (Vol 275) built a familyof solution blowing up if $n\geq 4$ and $\varepsilon$ small enough. Theyconjectured in their paper the existence of sign changing solutions which blowup and blow down at the same point. Here we give a confirmative answer byproving that our slightly subcritical problem has a solution with the shape ofsign changing bubbles concentrating on a stable critical point of the Robinfunction for $\varepsilon$ sufficiently small.

Stable critical point of the Robin function and bubbling phenomenon for a slightly subcritical elliptic problem