We consider the Brezis-Nirenberg problem: $$-Delta u =lambda u + |u|^{p-1}uqquad mbox{in},, Omega,quad u=0,, mbox{on},, partialOmega,$$ where $Omega$ is a smooth bounded domain in $R$, $Ngeq 3$, $p=rac{N+2}{N-2}$ and $lambda>0$. In this paper we prove that, if $Omega$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $lambda ightarrow lambda_1$, where $lambda_1=lambda_1(Omega)$ denotes the first eigenvalue of $-Delta$ on $Omega$, with zero Dirichlet boundary condition.
Sign-changing blowing-up solutions for the Brezis-Nirenberg problem in dimensions four and five / Iacopetti, Alessandro; Vaira, Giusi. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 2036-2145. - (2018). [10.2422/2036-2145.201602_003]
Sign-changing blowing-up solutions for the Brezis-Nirenberg problem in dimensions four and five
Iacopetti, Alessandro;Vaira, Giusi
2018
Abstract
We consider the Brezis-Nirenberg problem: $$-Delta u =lambda u + |u|^{p-1}uqquad mbox{in},, Omega,quad u=0,, mbox{on},, partialOmega,$$ where $Omega$ is a smooth bounded domain in $R$, $Ngeq 3$, $p=rac{N+2}{N-2}$ and $lambda>0$. In this paper we prove that, if $Omega$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $lambda ightarrow lambda_1$, where $lambda_1=lambda_1(Omega)$ denotes the first eigenvalue of $-Delta$ on $Omega$, with zero Dirichlet boundary condition.File | Dimensione | Formato | |
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