The Lazer-McKenna conjecture and a free boundary problem in two dimensions

We prove that certain super-linear elliptic equations in two dimensions have many solutions when the diffusion is small. We find these solutions by constructing solutions with many sharp peaks. In three or more dimensions, this has already been proved by the authors in 'Comm. Partial Differential Equations' 30 (2005) 1331-1358. However, in two dimensions, the problem is much more difficult because there is no limit problem in the whole space. Therefore, the proof is quite different, though still a reduction argument. A direct consequence of this result is that we give a positive answer to the Lazer-McKenna conjecture for some typical nonlinearities in two dimensions.