Lions' penalty condition for critical points of mountain pass type in semilinear elliptic problems

June 12th, 2007 at 15:15 in HG G43

Given by: Kyril Tintarev (Uppsala)

Abstract: In the concentration compactness framework of P.-L.-Lions, if $c$ is the constrained infimum value for a semilinear elliptic
problem without compactness and $c_\infty$ is the constrained infimum
for the correspondent problem at infinity,
then $c\le c_\infty$, while the strict inequality $c< c_\infty$ is a
general sufficient condition for existence of minima.
This penalty condition extends to the (unconstrained) mountain pass
levels in a variety of non-compact problems, where the functional(s) at
infinity are defined by limits with respect to translations, dilations or both. Particular attention will be given to asymptotically
selfsimilar nonlinearities oscillating about the critical stem $|u|^{2^*}$.