Abstract
In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.
Highlights
1 Introduction and main results The paper is devoted to the study of the following nonlinear sixth-order Hénon type elliptic equation:
The fourth-order Hénon type equation: 2u = |x|a|u|p–1u, in Rn studied by Hu [11]. He proved Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions. His proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence
Inspired by the ideas in [10, 13], our purpose in this paper is to prove the Liouville-type theorems in the class of stable solution and finite Morse index solution
Summary
Harrabi and Rahal [10] proved various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of Rn. Again, following [6, 9, 17], they established the standard integral estimates via stability property to derive the nonexistence results in the subcritical case by the use of Pohozaev identity. He proved Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Theorem 1.1 Let u ∈ C6(Rn) be a stable solution of (1.1) and 1 < p < pa(n, 6).
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