Liouville type theorems for Schrödinger systems

Abstract

We study positive solutions to the following higher order Schrödinger system with Dirichlet boundary conditions on a half space:

$$\left\{ \begin{gathered} ( - \Delta )^{\tfrac{\alpha } {2}} u(x) = u^{\beta _1 } (x)v^{\gamma _1 } (x), in R_ + ^n , \hfill \\ ( - \Delta )^{\tfrac{\alpha } {2}} u(x) = u^{\beta _2 } (x)v^{\gamma _2 } (x), in R_ + ^n , \hfill \\ u = \tfrac{{\partial u}} {{\partial x_n }} = \cdots = \tfrac{{\partial ^{\tfrac{\alpha } {2} - 1} u}} {{\partial x_n \tfrac{\alpha } {2} - 1}} = 0, on \partial R_ + ^n , \hfill \\ v = \tfrac{{\partial v}} {{\partial x_n }} = \cdots = \tfrac{{\partial ^{\tfrac{\alpha } {2} - 1} v}} {{\partial x_n \tfrac{\alpha } {2} - 1}} = 0, on \partial R_ + ^n , \hfill \\ \end{gathered} \right.$$

((0.1))

where α is any even number between 0 and n. This PDE system is closely related to the integral system

$$\left\{ {\begin{array}{*{20}c} {u(x) = \int_{R_ + ^n } {G(x,y)u^{\beta _1 } (y)v^{\gamma _1 } (y)dy,} } \\ {v(x) = \int_{R_ + ^n } {G(x,y)u^{\beta _2 } (y)v^{\gamma _2 } (y)dy,} } \\ \end{array} } \right.$$

((0.2))

where G is the corresponding Green’s function on the half space. More precisely, we show that every solution to (0.2) satisfies (0.1), and we believe that the converse is also true. We establish a Liouville type theorem — the non-existence of positive solutions to (0.2) under a very weak condition that u and v are only locally integrable. Some new ideas are involved in the proof, which can be applied to a system of more equations.