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Journal of the London Mathematical Society Advance Access published online on July 11, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn045
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© 2008 London Mathematical Society

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

E. N. Dancer

School of Mathematics and Statistics
University of Sydney
New South Wales 2006
Australia

Shusen Yan

School of Mathematics, Statistics and Computer Science
The University of New England
Armidale
New South Wales 2351
Australia
syan@turing.une.edu.au

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.

2000 Mathematics Subject Classification 35J65.

This work is partially supported by ARC.

Received December 5, 2007;
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