Sign-changing blowing-up solutions for supercritical Bahri-Coron's problem

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^n$, $n\ge 3$ with smooth boundary $\partial \Omega$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$-\Delta w=|w{|}^{\frac{4}{n-2}}w,w\in {\mathcal{D}}^{1,2}({\mathbb{R}}^n)$$ which has large number ($3n$) of kernels.

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References in corpus (2)

• Conformal approach to cylindrical DLA
• Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem