Monotonicity formula and $$\varepsilon $$ ε -regularity of stable solutions to supercritical problems and applications to finite Morse index solutions

Abstract

We prove a monotonicity formula and an -regularity theorem for stable solutions to a class of weighted supercritical semilinear elliptic equations. We then use them to study the behavior of finite Morse index solutions and obtain some sharp results, which improve those of Dancer et al. (J Differ Equ 250:3281-3310, 2011) and Du and Guo (Adv Differ Eqns 2013) and completely answer the questions left open there.