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Minimizers and symmetric minimizers for problems with critical Sobolev exponent
Stockholm University, Faculty of Science, Department of Mathematics.
2009 (English)In: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, ISSN 1230-3429, Vol. 34, no 2, 291-326 p.Article in journal (Other academic) Published
Abstract [en]

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces $\dkp(\real^N)$, where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding $\dkp(\real^N)\hookrightarrow L^{\qs} (\real^N,Q)$ when $Q$ is a non-negative, continuous, bounded function. However if $Q$ has certain symmetry properties then all minimizing sequences are relatively compact in the Sobolev space of appropriately symmetric functions. For $Q$ which does not have the required symmetry, we give a condition under which an equivalent norm in $\dkp(\real^N)$ exists so that all minimizing sequences are relatively compact. In fact we give an example of a $Q$ and an equivalent norm in $\dkp(\real^N)$ so that all minimizing sequences are relatively compact.

Place, publisher, year, edition, pages
2009. Vol. 34, no 2, 291-326 p.
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:su:diva-56489OAI: oai:DiVA.org:su-56489DiVA: diva2:411579
Available from: 2011-04-18 Created: 2011-04-18 Last updated: 2011-04-26Bibliographically approved
In thesis
1. Topics in Nonlinear Elliptic Differential Equations
Open this publication in new window or tab >>Topics in Nonlinear Elliptic Differential Equations
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we examine the existence of solutions to nonlinear elliptic partial differential equations via variational methods.

In Paper I we consider the existence of constrained minimizers which correspond to solutions of equations involving the iterated Laplacian, the iterated p-Laplacian and the critical Sobolev exponent. Particular attention is paid to problems with symmetries.

In Paper II we work on singular elliptic problems related to the Caffarelli-Kohn-Nirenberg-Lin inequality, which generalizes the Sobolev inequality. We prove the existence of solutions which break the symmetry of the underlying problem and have a prescribed number of nodal domains.

In Paper III we consider a different class of weighted problems related to the Caffarelli-Kohn-Nirenberg-Lin inequality. We establish the existence of at least one nontrivial solution. In the case $p=2$ (the iterated Laplacian) we show that there are infinitely many solutions.

In Paper IV we extend the results of Paper III concerning the existence of infinitely many solutions to the case $p\neq 2$ (the iterated p-Laplacian) and to a larger class of weights.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2011. 125 p.
Keyword
Concentration-compactness principle, critical Sobolev exponent, symmetric solutions of elliptic equations, degenerate elliptic equation, Nehari manifold, Caffarelli-Kohn-Nirenberg inequality, infinitely many solutions
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-56727 (URN)978-91-7447-279-0 (ISBN)
Public defence
2011-06-01, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Submitted.

Available from: 2011-05-11 Created: 2011-04-26 Last updated: 2013-04-02Bibliographically approved

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