Existence and nonexistence of sign-changing solutions to elliptic critical equations

Authors: MOKHLESS HAMMAMI, HOURIA ISMAIL

Abstract: We consider the nonlinear equation $ -\Delta u = | u|^{p-1}u -\varepsilon u \quad \mbox{in } \Omega , u =0 \quad \mbox{on } \partial \Omega ,$ where $\Omega $ is a smooth bounded domain in $\mathbb{R}^n$, $n \geq 4$, $ \varepsilon$ is a small positive parameter, and $p=(n+2)/(n-2)$. We study the existence of sign-changing solutions that concentrate at some points of the domain. We prove that this problem has no solutions with one positive and one negative bubble. Furthermore, for a family of solutions with exactly two positive bubbles and one negative bubble, we prove that the limits of the blow-up points satisfy a certain condition.

Keywords: Blow-up analysis, critical Sobolev exponent, sign-changing solutions

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