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

Abstract

In this paper, we deal with the boundary value problem −∆u = |u| 4/(n−2) u/[ln(e + |u|)] ε in a bounded smooth domain Ω in R n , n ≥ 3 with homogenous Dirichlet boundary condition. Here ε > 0. Clapp et al. in Journal of Diff. Eq. (Vol 275) built a family of solution blowing up if n ≥ 4 and ε 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 ε sufficiently small.