Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian

Self Cite

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.

Section: Proposition 3 (See Zhuo and LImentioning

confidence: 99%

Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian

Zhuo¹,

Li²

2019

Self Cite

“…It is worth mentioning that due to our nonlinear terms containing x and ∇u i , the non-degeneracy assumption in the present paper is just like (5) except replacing f i (u) by f i (x, u, ∇u i ). Recently, Zhuo-Chen-Cui-Yuan [38] showed the equivalence between the system (4) with m ≥ 2 and a corresponding integral system for 0 < α < n. Then once it's done, under the assumption (5), the rest immediately follows from the result in Chen-Li-Ou [14], where this integral equation has been well studied. However, both [38] and [13] need the assumption…”

mentioning

confidence: 96%

“…On the one hand, we extent the case α = 2 in Troy [36], Figueiredo [18] and Sirakov [35] to the fractional case 0 < α < 2. On the other hand, compared with the condition (6) in [38] for the problem in the whole space, we remove the assumptions that the nonnegativity of the derivatives ∂fi ∂ui and the homogeneity of the nonlinear terms, that is, our results include cooperative systems which is widely used for elliptic systems. Certainly, our results also include the supercritical case.…”

mentioning

confidence: 99%

Symmetry properties in systems of fractional Laplacian equations

Wu¹,

Xu²

2019

We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in R n. By using a direct method of moving planes, we show that u i (x) (i = 1, 2, • • • , m) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of ∂f i ∂u i. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.

“…In the literature, many new kinds of maximum principles have been developed [18,15,16,30,6]. Those maximum principles together with the moving plane methods [17,1,5] make it possible to work directly on the nonlocal equation.…”

mentioning

confidence: 99%

Direct method of moving planes for logarithmic Laplacian system in bounded domains

Liu¹

2018

Chen, Li and Li [Adv. Math., 308(2017), pp. 404-437] developed a direct method of moving planes for the fractional Laplacian. In this paper, we extend their method to the logarithmic Laplacian. We consider both the logarithmic equation and the system. To carry out the method, we establish two kinds of narrow region principle for the equation and the system separately. Then using these narrow region principles, we give the radial symmetry results for the solutions to semi-linear logarithmic Laplacian equations and systems on the ball.