Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space

Ran Zhuo; Fengquan Li; Boqiang Lv

doi:10.3934/cpaa.2014.13.977

Article Contents

2014, Volume 13, Issue 3: 977-990. Doi: 10.3934/cpaa.2014.13.977

This issue Previous Article Next Article Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space

Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space

1.
Department of Mathematical Sciences, Yeshiva University, New York, NY 10033
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024
3.
College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, Jiangxi 330063

Received: October 31, 2012

Revised: July 31, 2013

Published: April 2015

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Abstract

In this paper, we study the positive solutions for the following integral system: \begin{eqnarray} u(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_1}(y)v^{\gamma_1}(y)dy ,\\ v(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_2}(y)v^{\gamma_2}(y)dy, \end{eqnarray} where $0 < \alpha < n$ and $x^*=(x_1,\cdots,x_{n-1},-x_n)$ is the reflection of the point $x$ about the plane $R^{n-1}$, and $\beta_1, \gamma_1, \beta_2, \gamma_2 $ satisfy the condition$(f_1)$: \begin{eqnarray} 1 \leq \beta_1,\gamma_1,\beta_2,\gamma_2 \leq \frac{n+\alpha}{n-\alpha}\ \mbox{with}\ \beta_1+\gamma_1= \frac{n+\alpha}{n-\alpha}=\beta_2+\gamma_2, \beta_1\neq \beta_2, \gamma_1 \neq \gamma_2. \end{eqnarray}

This integral system is closely related to the PDE system with Navier boundary conditions, when $\alpha$ is a even number between $0$ and $n$, \begin{eqnarray} (- \Delta)^{\frac{\alpha}{2}}u(x)=u^{\beta_1}(x)v^{\gamma_1}(x), \mbox{in}\ R^n_+,\\ (- \Delta)^{\frac{\alpha}{2}}v(x)=u^{\beta_2}(x)v^{\gamma_2}(x), \mbox{in}\ R^n_+,\\ u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} u(x)=0,\mbox{on}\ \partial{R^n_+},\\ v(x)=-\Delta v(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} v(x)=0,\mbox{on}\ \partial{R^n_+}. \end{eqnarray}

More precisely, any solution of (1) multiplied by a constant satisfies (2). We use method of moving planes in integral forms introduced by Chen-Li-Ou to derive rotational symmetry, monotonicity, and non-existence of the positive solutions of (1) on the half space $R^n_+$.

Keywords:

Navier boundary conditions,
system of integral equations,
method of moving planes in integral forms,
Kelvin transforms,
symmetry,
monotonicity,
non-existence.

Mathematics Subject Classification: Primary: 31A10, 35B45; Secondary: 35B53, 35J91.

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