Critical point theory with applications to semilinear problems without compactness

• Main Author: Maad, Sara
• Format: Doctoral Thesis
• Language: English
• Published: Uppsala universitet, Matematiska institutionen 2002
• Subjects: Mathematics; MATEMATIK; MATHEMATICS
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-2061; http://nbn-resolving.de/urn:isbn:91-506-1557-2

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.