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Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 385, no 2, 721-736 p.Article in journal (Refereed) Published
Abstract [en]

We prove the existence of nontrivial critical points of the functional J(lambda)(u) = integral(RN)1/p(vertical bar vertical bar x vertical bar(-a del k)u vertical bar(p) - lambda h(x)vertical bar vertical bar x vertical bar(-(a+k))u vertical bar(p)) - 1/qQ(x)vertical bar vertical bar x vertical bar(-b)u vertical bar(q)dx, related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. As a consequence we obtain nontrivial solutions of the degenerate elliptic equation Delta(vertical bar x vertical bar(-ap)vertical bar Delta u vertical bar(p-2)Delta u) - lambda h(x)vertical bar x vertical bar(-(a+k)p)vertical bar u vertical bar(p-2)u = Q(x)vertical bar x vertical bar(-bq)vertical bar u vertical bar(q-2)u. We also show that when p = 2. J(lambda) has infinitely many critical points.

Place, publisher, year, edition, pages
2011. Vol. 385, no 2, 721-736 p.
Keyword [en]
Nehari manifold, Caffarelli-Kohn-Nirenberg inequality, Sign-changing weight function, Infinitely many solutions
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-56040DOI: 10.1016/j.jmaa.2011.07.005ISI: 000295062600012OAI: oai:DiVA.org:su-56040DiVA: diva2:408691
Note
1Available from: 2011-04-16 Created: 2011-04-05 Last updated: 2017-12-11Bibliographically 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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