The Hénon equation with a critical exponent under the Neumann boundary condition

Jaeyoung Byeon; Sangdon Jin

doi:10.3934/dcds.2018190

Article Contents

2018, Volume 38, Issue 9: 4353-4390. Doi: 10.3934/dcds.2018190

This issue Previous Article Traveling wave solutions for time periodic reaction-diffusion systems Next Article Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

The Hénon equation with a critical exponent under the Neumann boundary condition

Jaeyoung Byeon and
Sangdon Jin

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Received: August 2017

Revised: April 2018

Published: September 2018

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Abstract

For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Hénon equation with the homogeneous Neumann boundary condition

$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$

where $Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$ It is well known that for $α = 0,$ there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for $α > 0$ and its asymptotic behavior as the parameter $α$ approaches from below to a threshold $α_0 ∈ (0,∞]$ for existence of a least energy solution.

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Mathematics Subject Classification: Primary: 35B33, 35B40, 35J15, 35J25, 35J61.

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