Topics in Nonlinear Elliptic Differential Equations

Sammanfattning: 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 $p eq 2$ (the iterated p-Laplacian) and to a larger class of weights.