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A Supercritical Elliptic Problem in a Cylindrical Shell
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)In: Analysis and Topology in Nonlinear Differential Equations: A Tribute to Bernhard Ruf on the Occasion of his 60th Birthday / [ed] de Figueiredo et al, Basel: Birkhäuser , 2014, 233-242 p.Chapter in book (Refereed)
Abstract [en]

We consider the problem \[-\Delta u=\left\vert u\right\vert ^{p-2}u\text{ \ in }\Omega,\quad u=0\text{\ on }\partial\Omega,\]where $\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}%:0<a<\left\vert y\right\vert <b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2.$ Let$2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if$m=N-2$ or $N-1.$ We show that $2_{N,m}^{\ast}$ is the true critical exponent forthis problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ butthere are no such solutions if $p\geq2_{N,m}^{\ast}$.

Place, publisher, year, edition, pages
Basel: Birkhäuser , 2014. 233-242 p.
Series
Progress in Nonlinear Differential Equations and Their Applications, 85
Keyword [en]
Nonlinear elliptic equation, supercritical nonlinearity, unbounded domain
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-102354ISBN: 978-3-319-04213-8 (print)OAI: oai:DiVA.org:su-102354DiVA: diva2:709774
Funder
Swedish Research Council
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-08Bibliographically approved

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