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

Abstract

For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Hénon equation with the homogeneous Neumann boundary condition \begin{document}$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$ \end{document} 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.

Locations

• Discrete and Continuous Dynamical Systems - View - PDF