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Topics in Nonlinear Elliptic Differential Equations
Stockholm University, Faculty of Science, Department of Mathematics.
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 [en]
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: urn:nbn:se:su:diva-56727ISBN: 978-91-7447-279-0 (print)OAI: oai:DiVA.org:su-56727DiVA: diva2:412818
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At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript.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
List of papers
1. Infinitely many solutions for some singular elliptic problems
Open this publication in new window or tab >>Infinitely many solutions for some singular elliptic problems
2013 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 33, no 1, 321-333 p.Article in journal (Refereed) Published
Abstract [en]

We prove the existence of an unbounded sequence of critical points of the functional J(lambda) (u) = 1/p integral(RN) parallel to x vertical bar(-a)del(k)u vertical bar(p) - lambda h(x)parallel to x vertical bar(-(a+k))u vertical bar(p) - 1/q integral(RN) Q(x)parallel to x vertical bar(-b) u vertical bar(q) related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume Q <= 0 at 0 and infinity and consider two essentially different cases: h equivalent to 1 and h in a certain weighted Lebesgue space.

Keyword
Nehari manifold, Caffarelli-Kohn-Nirenberg inequality, sign-changing weight function, infinitely many solutions
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-82151 (URN)10.3934/dcds.2013.33.321 (DOI)000309286500022 ()
Note

AuthorCount:2;

Available from: 2012-11-09 Created: 2012-11-08 Last updated: 2017-12-07Bibliographically approved
2. Sign-changing and symmetry-breaking solutions to singular problems
Open this publication in new window or tab >>Sign-changing and symmetry-breaking solutions to singular problems
2012 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 57, no 11, 1191-1208 p.Article in journal (Refereed) Published
Abstract [en]

We consider the degenerate elliptic equation -div(vertical bar x vertical bar(-ap) vertical bar del(u)vertical bar(p-2) del(u)) - lambda vertical bar x vertical bar(-p(a+1))vertical bar u vertical bar(p-2)u = vertical bar x vertical bar(-bq) vertical bar u vertical bar(q-2)u in R-N related to the Caffarelli-Kohn-Nirenberg inequality. We show that it possesses infinitely many solutions which are sign-changing and nonradial. The solutions are obtained by constrained minimization on subspaces consisting of functions which have certain prescribed symmetry properties. We also extend these results to higher order equations.

Keyword
concentration-compactness principle, degenerate elliptic equation, sign-changing solution, symmetry-breaking solution
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-82978 (URN)10.1080/17476933.2010.504849 (DOI)000310132900004 ()
Note

AuthorCount:2;

Available from: 2012-12-07 Created: 2012-12-03 Last updated: 2017-12-07Bibliographically approved
3. Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type
Open this publication in new window or tab >>Higher order singular problems of Caffarelli-Kohn-Nirenberg-Lin type
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.

Keyword
Nehari manifold, Caffarelli-Kohn-Nirenberg inequality, Sign-changing weight function, Infinitely many solutions
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-56040 (URN)10.1016/j.jmaa.2011.07.005 (DOI)000295062600012 ()
Note
1Available from: 2011-04-16 Created: 2011-04-05 Last updated: 2017-12-11Bibliographically approved
4. Minimizers and symmetric minimizers for problems with critical Sobolev exponent
Open this publication in new window or tab >>Minimizers and symmetric minimizers for problems with critical Sobolev exponent
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.

National Category
Natural Sciences
Identifiers
urn:nbn:se:su:diva-56489 (URN)
Available from: 2011-04-18 Created: 2011-04-18 Last updated: 2011-04-26Bibliographically approved

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