Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian

Abstract

In this paper, we consider the following Schrödinger systems involving pseudo-differential operator in R n (−∆) α 2 u(x) = u β 1 (x)v τ 1 (x), in R n , (−∆) γ 2 v(x) = u β 2 (x)v τ 2 (x), in R n , (1) where α and γ are any number between 0 and 2, α does not identically equal to γ. We employ a direct method of moving planes to partial differential equations (PDEs) (1). Instead of using the Caffarelli-Silvestre's extension method and the method of moving planes in integral forms, we directly apply the method of moving planes to the nonlocal fractional order pseudo-differential system. We obtained radial symmetry in the critical case and non-existence in the subcritical case for positive solutions. In the proof, combining a new approach and the integral definition of the fractional Laplacian, we derive the key tools, which are needed in the method of moving planes, such as, narrow region principle, decay at infinity. The new idea may hopefully be applied to many other problems.

On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

“…Furthermore, also relevant to this study is the focus of qualitative research in symmetric domains to provide the symmetries of solutions (mainly, working with kinematical and dynamical algebras of Schrödinger type equations). A combination of suitable iteration methods, maximum principle and method of moving planes, is useful to detect symmetries of positive solutions and nonexistence results (see, for example, [3]). We also mention the recent works of Peng-Zhao [4] (global existence and blow-up of solutions) and Hoshino [5] (asymptotic behavior of solutions).…”

Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

“…The fractional and nonlocal operators of elliptic type due to concrete real world have been applied in finance, thin obstacle problem, optimization, quasi-geostrophic flow, etc. In recent years, there have been tremendous interests in developing the problems related to fractional Laplacian (see in [1][2][3][4][5] and the reference therein). Fractional p-Laplacian is a generalization of the fractional Laplacian, but there are few interesting results about these nonlinear equations involving fractional p-Laplacian.…”

“…When = 2, the fractional p-Laplacian (−Δ) becomes the well-known fractional Laplacian operator (−Δ) . In [2], 2 Advances in Mathematical Physics the authors considered the following system involving fractional Laplacian on the whole space:…”

“…Assume that, for ≥ 0, , are nonnegative continuous functions satisfying the following: 2 ) and ( ) = 2 ( +2 )/( −2 ) . Motivated by the ideas of [2,6], our main concern in this paper is to study the following nonlinear system involving fractional p-Laplacian in a unit ball:…”

“…Remark 2. In [3,9], the authors introduced the direct method of moving planes, respectively, to prove symmetry, monotonicity, and nonexistence of solutions to various differential equations on the whole space and on a half space, etc., such as ones involving fractional Laplacian in [2,3], fractional p-Laplacian in [6], and fully nonlinear operators in [4,8]. This direct method of moving planes has some advantages, which overcome the necessary of imposing extra assumption on the solutions when using the extension method (see also [10]) or the equivalent integral equation method (see also [11]).…”