Topics in Nonlinear Elliptic Differential Equations
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Concentration-compactness principle, critical Sobolev exponent, symmetric solutions of elliptic equations, degenerate elliptic equation, Nehari manifold, Caffarelli-Kohn-Nirenberg inequality, infinitely many solutions
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-56727ISBN: 978-91-7447-279-0OAI: oai:DiVA.org:su-56727DiVA: diva2:412818
2011-06-01, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Bartsch, Thomas, Professor
Szulkin, Andrzej, Professor
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript.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.2011-05-112011-04-262013-04-02Bibliographically approved
List of papers