Liouville-type theorems and existence results for stable solutions to weighted Lane–Emden equations

Abstract

We devote this paper to proving non-existence and existence of stable solutions to weighted Lane-Emden equations on the Euclidean space ℝN, N ⩾ 2. We first prove some new Liouville-type theorems for stable solutions which recover and considerably improve upon the known results. In particular, our approach applies to various weighted equations, which naturally appear in many applications, but that are not covered by the existing literature. A typical example is provided by the well-know Matukuma's equation. We also prove an existence result for positive, bounded and stable solutions to a large family of weighted Lane–Emden equations, which indicates that our Liouville-type theorems are somehow sharp.

Liouville type theorems for solutions of the weighted fractional Lane-Emden system

“…Many other papers studied stable solutions of (1.10) with s = 1, see for instance [5,8,28,3,24]. In particular, Farina-Hasegawa [18] proved Liouville type results for stable solutions to (1.10) with s = 1 and a larger class of weights h which cover many existing results. Davila-Dupaigne-Wei [9] examined the equation (1.10) with h ≡ 1, and classified finite Morse index solutions in the autonomous case.…”

Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

“…Many other papers studied stable solutions of (10) with s = 1, see for instance [5,10,28,3,24]. In particular, Farina-Hasegawa [18] proved Liouville type results for stable solutions to (10) with s = 1 and a larger class of weights h which cover many existing results.…”