Liouville theorems for the weighted Lane-Emden equation with finite Morse indices

Abstract

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: −normalΔu=f(|x|)|u|p−1u,2emx∈double-struckRN and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: {array−normalΔu=f(|x|)|u|p−1u,arrayx∈R+N,array∂u∂ν=g(|x|)|u|−1u,arrayx∈∂R+N, where f(|x|) and g(|x|) are the radial and continuously differential functions, double-struckR+N={x=(x′,xN)∈double-struckRN−1×double-struckR+} is an upper half space in double-struckRN, and ∂double-struckR+N={x=(x′,0),3.0235ptx′∈double-struckRN−1}. Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems and under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.

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

“…Note that the weight (A) satisfies condition (W 2 ) with = −1. Also note that the above theorem 1.1 recovers and considerably improves upon the Liouville-type theorems established in [4,11], where stable solutions were assumed to be positive and/or bounded.…”

“…In [7,12,15], the authors also treated stable solutions to (W) with W = |x| . In [4], Chen and Wang obtained Liouville theorems on non-trivial, bounded and stable solutions to (W) with a radial weight W satisfying some precise decay and monotonicity assumptions. Furthermore, Hajlaoui et al [11] proved the non-existence of positive and stable solutions to (W), where the weight W satisfies the following assumption:…”

“…However, there are some interesting weight functions W that do not satisfy ( * ) and thus they are not covered by the previous works [4,11]. Two typical examples are provided by the following cases:…”

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

“…Finally, the same resultat is proved to be true without the local boundedness assumption by Wang-Ye [38]. 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.…”

Stability and separation property of radial solutions to semilinear elliptic equations

“…Hence, p JL (N, ) is also the critical exponent on the existence of nontrivial stable solutions to (1) in the case of K(|x|) = |x| . Moreover, nonexistence results remain valid for a general K satisfying C 1 r ≤ K(r) ≤ C 2 r with some constants C 1 , C 2 > 0 (see Remark 1.8 (i) in [10]), and the Liouville property of stable solutions was also obtained for the equation 1replacing u p by e u or a generalized nonlinear term f (u) ( [9,12,14,15,16,30]) and for the elliptic systems ( [20]). However, the existence of non-trivial stable solutions was shown for only the case K(|x|) = |x| in [10,13], and the proofs of them depend heavily on the fact that K is precisely given.…”