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 -Δu = |u|4/(n-2)u/[ln (e+|u|)]ε in a bounded smoothdomain Ω in ℝn, n ≥ 3 with homogenousDirichlet boundary condition. Here ε > 0. Clapp et al. in Journalof 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. Mathematics Subject Classification 2000: 35J20, 35J60.