High energy sign-changing solutions for Coron's problem

Abstract

We study the existence of sign changing solutions to the following problem $$(P)\{\begin{array}{llc}\Delta u+|u{|}^{p-1}u=0& \mathrm{in}{\Omega }_{\epsilon };u=0& \mathrm{on}\partial {\Omega }_{\epsilon },\end{array}$$ where $p=\frac{n+2}{n-2}$ is the critical Sobolev exponent and ${\Omega }_{\epsilon }$ is a bounded smooth domain in ${\mathcal{R}}^n$, $n\ge 3$, with the form ${\Omega }_{\epsilon }=\Omega \backslash B(0,\epsilon )$ with $\Omega$ a smooth bounded domain containing the origin $0$ and $B(0,\epsilon )$ the ball centered at the origin with radius $\epsilon >0$. We construct a new type of sign-changing solutions with high energy to problem $(P)$, when the parameter $\epsilon$ is small enough.

39 pages

References in corpus (1)

• Nondegeneracy of Nonradial Nodal Solutions to Yamabe Problem