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Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group
Department of Mathematics, University of Virginia.
2005 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 116, no 3, 357-384 p.Article in journal (Refereed) Published
Abstract [en]

We study the semilinear equationwhere  is the Heisenberg Laplacian and  is the Heisenberg group. The function f  C2(×) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of  consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S12(), which is the Heisenberg analogue of the Sobolev space W1,2(N).

Place, publisher, year, edition, pages
2005. Vol. 116, no 3, 357-384 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-89884DOI: 10.1007/s00229-004-0534-1OAI: oai:DiVA.org:uu-89884DiVA: diva2:161716
Available from: 2002-05-06 Created: 2002-05-06 Last updated: 2017-12-14Bibliographically approved
In thesis
1. Critical point theory with applications to semilinear problems without compactness
Open this publication in new window or tab >>Critical point theory with applications to semilinear problems without compactness
2002 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of four papers which all regard the study of critical point theory and its applications to boundary value problems of semilinear elliptic equations. More specifically, let Ω be a domain, and consider a boundary value problem of the form -L u + u = f(x,u) in Ω, and with the boundary condition u=0. L denotes a linear differential operator of second order, and in the papers, it is either the classical Laplacian or the Heisenberg Laplacian defined on the Heisenberg group. The function f is subject to some regularity and growth conditions.

Paper I contains an abstract result about nonlinear eigenvalue problems. We give an application to the given equation when L is the classical Laplacian, Ω is a bounded domain,

and f is odd in the u variable.

In paper II, we study a similar equation, but with Ω being an unbounded domain of N-dimensional Euclidean space. We give a condition on Ω for which the equation has infinitely many weak solutions.

In papers III and IV we work on the Heisenberg group instead of Euclidean space, and with L being the Heisenberg Laplacian.

In paper III, we study a similar problem as in paper II, and give a condition on a subset Ω of the Heisenberg group for which the given equation has infinitely many solutions. Although the condition on Ω is directly transferred from the Euclidean to the Heisenberg group setting, it turns out that the condition is easier to fulfil in the Heisenberg group than in Euclidean space.

In paper IV, we are still on the Heisenberg group, Ω is the whole group, and we study the equation when f is periodic in the x variable. The main result is that also in this case, the equation has infinitely many solutions.

Place, publisher, year, edition, pages
Uppsala: Avdelningen för matematik, 2002. 75 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 23
Keyword
Mathematics, MATEMATIK
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-2061 (URN)91-506-1557-2 (ISBN)
Public defence
2002-05-29, Room 247, Building 2, Polacksbacken, Uppsala, 13:15
Opponent
Available from: 2002-05-06 Created: 2002-05-06 Last updated: 2012-07-26Bibliographically approved

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