Liouville type theorem for critical order Hénon-Lane-Emden type equations on a half space and its applications

Abstract

In this paper, we are concerned with the critical order Hénon-Lane-Emden type equations with Navier boundary condition on a half space ${\mathbb{R}}_{+}^n$: \begin{equation}\label{NPDE0}\\\begin{cases} (-Δ)^{\frac{n}{2}} u(x)=f(x,u(x)),\ u(x)\geq0,\ x\in\mathbb{R}^{n}_+, \\ u=(-Δ)u = \cdots = (-Δ)^{\frac{n}{2}-1}u = 0,\ x\in\partial\mathbb{R}^{n}_+, \end{cases}\end{equation} where $u\in C^n({\mathbb{R}}_{+}^n)\cap C^{n-2}(\overline{{\mathbb{R}}_{+}^n})$ and $n\ge 2$ is even. We first consider the typical case $f(x,u)=|x{|}^a{u}^p$ with $0\le a<\infty$ and $1<p<\infty$. We prove the super poly-harmonic properties and establish the equivalence between (0.1) and the corresponding integral equations \begin{equation}\label{IE0} u(x)=\int_{\mathbb{R}^{n}_+}G(x,y)f(y,u(y))dy, \end{equation} where $G(x,y)$ denotes the Green's function for $(-\Delta {)}^{\frac{n}{2}}$ on ${\mathbb{R}}_{+}^n$ with Navier boundary conditions. Then, we establish Liouville theorem for (0.2) via ``the method of scaling spheres" developed initially in \cite{DQ0} by Dai and Qin, and hence we obtain the Liouville theorem for (0.1) on ${\mathbb{R}}_{+}^n$. As an application of the Liouville theorem on ${\mathbb{R}}_{+}^n$ (Theorem 1.6) and Liouville theorems in ${\mathbb{R}}^n$ established in Chen, Dai and Qin [4] for $n\ge 4$ and Bidaut-Véron and Giacomini [1] for $n=2$, we derive a priori estimates and existence of positive solutions to critical order Lane-Emden equations in bounded domains for all $n\ge 2$ and $1<p<\infty$. Extensions to IEs and PDEs with general nonlinearities $f(x,u)$ are also included.

28 pages. arXiv admin note: text overlap with arXiv:1810.02752

References in corpus (5)

• Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres
• Symmetry and Asymmetry: The Method of Moving Spheres
• Nonradial solutions for the Hénon equation in $R^N$
• Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy-Hénon equations in ${\mathbb{R}}^n$
• Liouville type theorems, a priori estimates and existence of solutions for non-critical higher order Lane-Emden-Hardy equations

Cited by in corpus (1)

• Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains