On the uniqueness of positive solutions of the Lane-Emden problem in planar domains

The question of the uniqueness for the positive solutions of the Lane-Emden problem arose since the famous symmetry result by Gidas, Ni, Nirenberg (1979),  which implies uniqueness when the domain is a ball. A conjecture on the uniqueness in any convex domain was then formulated during the eighties, but only partial answers have been given so far.  In this talk we will describe recent results obtained in dim=2 about the asymptotic behavior of positive solutions, their non-degeneracy and Morse index computation. In particular we will show that the uniqueness conjecture in convex domains  is true in the planar case and for p large enough.

This is from joint works with Franesca De Marchis, Massimo Grossi, Filomena Pacella (University Sapienza of Roma, Italy), Peng Luo and Shusen Yan (Wuhan Normal University, China).