Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian

Abstract

In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: where . When Ω is the unit ball or , we prove that the solutions are radially symmetric and decreasing. When Ω is the parabolic domain on , we prove that the solutions are increasing. Furthermore, if Ω is the , then we also derive the nonexistence of positive solutions to the system on the half-space. We assume that the nonlinear terms f, g and the solutions u, v satisfy some amenable conditions in different cases.

Symmetry and Monotonicity of Positive Solutions to Schrödinger Systems with Fractional $p$-Laplacian

“…Remark 2.2. Compared with the narrow region principle for the Schrödinger system with fractional Laplace equations in [21], here we need to impose the extra assumption that there exist y 0 , y 1 ∈ Σ λ such that U λ (y 0 ) > 0 and V λ (y 1 ) > 0, respectively. As a matter of fact, this condition is automatically satisfied for (1.5), (1.8), (1.11) and (1.12).…”

A direct method of moving planes for fully nonlinear nonlocal operators and applications

“…In fact, these two methods have been applied successfully to study equations involving nonlocal fractional operators (−∆) s (0 < s < 1), and a series of fruitful results have been achieved(see [4,16,20,21,26,27,28,37,43,45] and the references therein). There also got a series of fruitful results for fractional elliptic systems(see [11,39,41,42,46]) by moving planes. The disadvantage of the above mentioned approaches is that one need to impose some additional conditions on the solutions.…”