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



Brezis-Nirenberg result compared to abstract bifurcation theory 
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 $\Omega$.


