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

2023/11/23 by Fourti, Habib, Ghoudi, Rabeh · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.14002

Abstract

In this paper, we deal with the boundary value problem -Δu= |u|^4/(n-2)u/[ln (e+|u|)]^ε in a bounded smooth domain Ω in ℝⁿ, 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.

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• Non existence of solutions for a slightly super-critical elliptic problem with non-power nonlinearity

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