<p>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.</p><p>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, </p><p>and f is odd in the u variable. </p><p>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. </p><p>In papers III and IV we work on the Heisenberg group instead of Euclidean space, and with L being the Heisenberg Laplacian. </p><p>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. </p><p>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. </p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:uu-2061 |
| Date | January 2002 |
| Creators | Maad, Sara |
| Publisher | Uppsala University, Mathematics, Uppsala : Avdelningen för matematik |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, text |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 23 |
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