×

Periodic solutions of fractional Laplace equations: least period, axial symmetry and limit. (English) Zbl 1519.34033

This paper is concerned with the following fractional Laplace equation \[ (-\partial_{xx})^{s}u(x) + F'(u(x)) = 0, \quad x\in \mathbb{R}, \] where \((-\partial_{xx})^{s}\), \(0<s<1\), denotes the usual fractional Laplacian; \(F\) is a smooth double-well potential satisfying \[ \left\{ \begin{array}{ll} F(1)=F(-1)=0< F(u)\text{ for all }-1<u<1;\\ F'(1)=F'(-1)=0 \end{array} \right. \] and \(F\) is nondecreasing in \((-1,0)\) and nonincreasing in \((0,1)\). The authors show that the value of least positive period is \(\frac{2\pi}{(-F''(0))^{1/(25)}}\). The axial symmetry of odd periodic solutions is obtained by moving plane method. The authors also prove that odd periodic solutions \(u_T(x)\) converge to a layer solution of the same equation as periods \(T\rightarrow +\infty\).
Reviewer: Minghe Pei (Jilin)

MSC:

34C25 Periodic solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrodinger equation, Nonlinear Anal. 120 (2015), 262-284. · Zbl 1320.35316
[2] V. Ambrosio, Periodic solutions for the non-local operator \((−\Delta+m^2)^s −m^{2s}\) with \(m \geq 0\), Topol. Methods Nonlinear Anal. 49 (2017), 75-103. · Zbl 1420.35452
[3] V. Ambrosio and G. Molica Bisci, Periodic solutions for nonlocal fractional equations, Commun. Pure Appl. Anal. 16 (2017), 331-344. · Zbl 1362.35317
[4] B. Barrios, J. Garcia-Melian and A. Quaas, Periodic solutions for the onedimensional fractional Laplacian, J. Differential Equations 267 (2019), 5258-5289. · Zbl 1430.35246
[5] X. Cabre and Y. Sire, Nonlinear equations for fractional Laplacians I. Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincare Anal. Non Lineaire 31 (2014), 23-53. · Zbl 1286.35248
[6] X. Cabre and Y. Sire, Nonlinear equations for fractional Laplacians II. Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367 (2015), 911-941. · Zbl 1317.35280
[7] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260. · Zbl 1143.26002
[8] W.X. Chen, C.M. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math. 308 (2017), 404-437. · Zbl 1362.35320
[9] Y.X. Cui and Z.Q. Wang, Multiple periodic solutions of a class of fractional Laplacian equations, Adv. Nonlinear Stud. 21 (2021), 41-56. · Zbl 1487.35011
[10] A. DelaTorre, M. del Pino, M.d.M. Gonzalez and J.C. Wei, Delaunay-type singular solutions for the fractional Yamabe problem, Math. Ann. 369 (2017), 597-626. · Zbl 1378.35321
[11] Z.R. Du and C.F. Gui, Further study on periodic solutions of elliptic equations with a fractional Laplacian, Nonlinear Anal. 193 (2020), Ariticle ID 111417. · Zbl 1436.35172
[12] Z.R. Du and C.F. Gui, Periodic solutions of Allen-Cahn system with the fractional laplacian, Nonlinear Anal. 201 (2020), Ariticle ID 112061. · Zbl 1455.34042
[13] E. Fabes, D. Jerison and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst Fourier (Grenoble) 32 (1982), 151-182. · Zbl 0488.35034
[14] E.B. Fabes, C.E. Kenig and R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), 77-116. · Zbl 0498.35042
[15] Z.P. Feng and Z.R. Du, Periodic solutions of non-autonomous Allen-Cahn equations involving fractional Laplacian, Adv. Nonlinear Stud. 20 (2020), 725-737. · Zbl 1445.35164
[16] Z.P. Feng and Z.R. Du, Multiple periodic solutions of Allen-Cahn system involving fractional Laplacian, preprint.
[17] C.F. Gui, J. Zhang and Z.R. Du, Periodic solutions of a semilinear elliptic equation with a fractional Laplacian, J. Fixed Point Theory Appl. 19 (2017), 363-373. · Zbl 1366.35042
[18] Y. Hu, Layer solutions for a class of semilinear elliptic equations involving fractional Laplacians, Bound. Value Probl. 2014 (2014), Article ID 41. · Zbl 1317.35282
[19] G. Palatucci, O. Savin and E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. 192 (2013), 673-718. · Zbl 1278.82022
[20] L. Roncal and P.R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. 18 (2016), 1550033, 26 pp. · Zbl 1383.35246
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.