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

In this paper, we deal with the boundary value problem \(-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon \) $-\mathrm{\Delta }u={|u|}^{4/(n-2)}u/[ln(e+{|u|)]}^{\epsilon }$ in a bounded smooth domain \( \Omega \) $\mathrm{\Omega }$ in \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , \(n\ge 3\) $n\ge 3$ with homogenous Dirichlet boundary condition. Here \(\varepsilon >0\) $\epsilon >0$ . Clapp et al. (J Differ Equ 275:418–446, 2021) built a family of solution blowing up if \(n\ge 4\) $n\ge 4$ and \(\varepsilon \) $\epsilon$ small enough. They conjectured in their paper the existence of sign changing solutions which blow up and blow down at the same point. Here we give a confirmative answer by proving that our slightly subcritical problem has a solution with the shape of sign changing bubbles concentrating on a stable critical point of the Robin function for \(\varepsilon \) $\epsilon$ sufficiently small.