Question

Dear Mathoverflow'ers,

I am interested in the following equation:

$-\Delta u = u^{p-1} + \lambda u$ in $ \Omega$ with $ u=0 $ on $ \partial \Omega$.

1) My question is related to the Brezis-Nirenberg result from 1983 which states (and I am probably slightly off here) that when $ p=2^*$ (the critical Sobolev exponent) that there is a positive solution for certain values of $ \lambda$ and they can give an optimal range of $ \lambda$ (in certain cases) where one has a positive solution.

2) I believe that at the time it was very suprising that the addition of this linear term could restore compactness in the critical imbedding and recover a positive solution.

3) In another direction one can use some abstract bifurcation theory (i believe it would be the results of Crandall-Rabinowitz from the mid seventies) to show there is a positive solution for any value of $ p>2$ (as large as one likes) provided $ \lambda$ was sufficiently close (and to the left) of the first eigenvalue of $ -\Delta$.

My question is that since 3) was already well known why was 1) so suprising?
I realize that this is somewhat of an ill formed question and may not be suitable for mathoverflow.

thanks Greg

Answer 1

There are several reasons why the work of Brezis-Nirenberg was surprising.

1. First, it goes beyond a small range of $\lambda$'s that one would obtain from bifurcation theory.

2. The existence of positive solutions is highly dependent on the geometry and topology of $\Omega$.