Event Details
Differential equations seminar
Speaker: Juan Silverio (Universidad de Granada)
ABSTRACT: In this preliminary work we will concern about the "Finite Morse Index" scenario (instead of the usual stability one) of the following long-standing conjecture: "Let u be a (compactly supported) weak stable solution of the general non-linear Poisson equation and assume that the non-linearity is positive, non-decreasing, convex, and superlinear at +∞, and let n<10. Then u is bounded." Recently, Cabré, Figalli, Ros-Oton and Serra end up a complete proof in the classical stability setting: W^{1,2}-stable solutions are universally bounded for n<10 (and therefore smooth by classical elliptic regularity theory); namely they are bounded in terms only of their L^1 norm, with a constant that is independent of the non-linearity. This conjecture is in a sense equivalent to another problem stated before by Brezis: Is it possible to prove that the extremal solution of the so-called Gelfand problem is smooth at least in low dimensions?. We will take this as the starting point to reproduce our new results recovering the existing results by S. Villegas in the (semi-)stable case. Using some estimates on the "size" of the local stability behavior of finite Morse Index solutions we provide a uniform a priori bound and some pointwise estimates of that solutions that partially answer positively the long standing conjecture in this more general setting.
