Solutions to the nonlinear Schr\xf6dinger systems involving the fractional Laplacian

Meng Qu,Liu Yang

doi:10.1186/s13660-018-1874-9

Abstract

In this paper, we consider the following nonlinear Schrödinger system involving the fractional Laplacian operator: \t\t\t{(−Δ)α2u+au=f(v),(−Δ)β2v+bv=g(u),on Ω⫅Rn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta)^{\\frac{\\alpha}{2}}u+au=f(v), \\\\ (-\\Delta)^{\\frac{\\beta}{2}}v+bv=g(u), \\end{cases}\\displaystyle \\quad \\text{on } \\Omega\\subseteqq \\mathbb{R}^{n}, $$\\end{document} where a,bgeq0. When Ω is the unit ball or mathbb{R}^{n}, we prove that the solutions (u,v) are radially symmetric and decreasing. When Ω is the parabolic domain on mathbb{R}^{n}, we prove that the solutions (u,v) are increasing. Furthermore, if Ω is the mathbb{R}^{n}_{+}, 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.

Highlights

This paper is mainly devoted to investigating the properties of the solutions of the following system involving the fractional Laplacian operators: ⎧ ⎨(–) 2 u + au = f (v), ⎩(– β) 2 v + bv = g(u), for some a, b ≥ 0, (1.1)with α u(x) – u(y)(– ) 2 u(x) = Cn,α P
Chen, Li, and Li [12] developed a new method that can handle directly these nonlocal operators. They used this property to develop some techniques needed in the direct method of moving planes in the whole space Rn and the upper half-space Rn+, such as the narrow region principle and decay at infinity
We consider the nonexistence of positive solutions to system (1.1) in the half-space

Summary

Introduction

This paper is mainly devoted to investigating the properties of the solutions of the following system involving the fractional Laplacian operators:. Chen, Li, and Li [12] developed a new method that can handle directly these nonlocal operators They used this property to develop some techniques needed in the direct method of moving planes in the whole space Rn and the upper half-space Rn+, such as the narrow region principle and decay at infinity. Theorem 1.1 Let u ∈ C(B) ∩ Cl1o,c1(B) and v ∈ C(B) ∩ Cl1o,c1(B) be positive solutions of the system. The parabolic domain on Rn. Theorem 1.2 Let u ∈ Lα(Rn) ∩ Cl1o,c1( ) and v ∈ Lβ (Rn) ∩ Cl1o,c1( ) be positive solutions of the system. U(x) and v(x) are radially symmetric and decreasing about some point x0 in Rn. we consider the nonexistence of positive solutions to system (1.1) in the half-space.

Since we can choose l small enough and

Choose a point in c λ