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 \(-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon \) in a bounded smooth domain \( \Omega \) in \({\mathbb {R}}^n\), \(n\ge 3\) with homogenous Dirichlet boundary condition. Here \(\varepsilon >0\). Clapp et al. (J Differ Equ 275:418–446, 2021) built a family of solution blowing up if \(n\ge 4\) and \(\varepsilon \) 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 \(\varepsilon \) sufficiently small.