Abstract
Abstract Let N ≥ 2 {N\geq 2} and 1 < p < ( N + 2 ) / ( N - 2 ) + {1<p<(N+2)/(N-2)_{+}} . Consider the Lane–Emden equation Δ u + u p = 0 {\Delta u+u^{p}=0} in ℝ N {\mathbb{R}^{N}} and recall the classical Liouville type theorem: if u is a non-negative classical solution of the Lane–Emden equation, then u ≡ 0 {u\equiv 0} . The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of Serrin and Zou, originally used for the Lane–Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type theorem in all dimensions.
Highlights
We revisit the Liouville property of non-negative entire solutions for the Lane-Emden equation∆u + up = 0, x ∈ Ω, (1.1)where N ≥ 2, 1 < p < (N + 2)/(N − 2)+ and Ω ⊆ RN
When Ω = RN and p = (N + 2)/(N − 2), the solvability of the Lane-Emden equation is related to a sharp Sobolev inequality and the classical Yamabe problem [LP87]
Equation (1.1) is the “blow-up” equation associated with a family of second-order elliptic equations with Dirichlet data
Summary
We revisit the Liouville property of non-negative entire solutions for the Lane-Emden equation. The proofs of both parts were simplified considerably in [CL91] with the aid of the Kelvin transform and the method of moving planes We refer those who are interested to [FAR07] and the references therein for more classification and Liouville type theorems for the Lane-Emden equation with respect to various types of solutions, e.g., stable, finite Morse index, radial, or sign-changing, etc. Obtaining an analogous non-existence theorem for this system, proved rather difficult It remains an open problem and is commonly known as the Lane-Emden conjecture. In some of the previous papers utilizing Serrin and Zou’s approach, the resulting Liouville type theorems were proved under a boundedness assumption on solutions. The method has been further developed recently and some notable papers are [AYZ14, CHL16, COW14, QS12]
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