# American Institute of Mathematical Sciences

January  2016, 36(1): 499-508. doi: 10.3934/dcds.2016.36.499

## Radial sign-changing solution for fractional Schrödinger equation

 1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, China

Received  January 2014 Revised  December 2014 Published  June 2015

We study the existence of a radial sign-changing solution for the stationary fractional Schrödinger equation $$\nonumber (-\Delta)^\alpha u + u=|u|^{p-1}u,\ x\in \mathbb{R}^N, N\geq 2.~~~~~~~~~~~~~{\rm (FS)}$$ where $\alpha\in (0,1)$ and $p\in(1,2_\alpha^*-1)$, $2_\alpha^*=\frac{2N}{N-2\alpha}$. By using Brouwer degree theory and variational method, we prove that there exists a radial sign-changing solution of (FS).
Citation: Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499
##### References:
 [1] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42. doi: 10.1081/PDE-120028842. [2] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005. [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556. [4] X. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494. doi: 10.1088/0951-7715/26/2/479. [5] P. Felmer, A. Quaas and J. G. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. [6] R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, arXiv:1302.2652. [7] B. L. Guo and D. W. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp. doi: 10.1063/1.4746806. [8] C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059. [9] N. Laskin, Fractional Schrödinger equations, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [10] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6. [11] C. Miranda, Un' osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7. [12] E. S. Noussair and J. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1255-1271. doi: 10.1512/iumj.1997.46.1401.

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##### References:
 [1] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42. doi: 10.1081/PDE-120028842. [2] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281. doi: 10.1016/j.anihpc.2004.07.005. [3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556. [4] X. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494. doi: 10.1088/0951-7715/26/2/479. [5] P. Felmer, A. Quaas and J. G. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh, Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. [6] R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, arXiv:1302.2652. [7] B. L. Guo and D. W. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys., 53 (2012), 083702, 15pp. doi: 10.1063/1.4746806. [8] C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986), 803-835. doi: 10.1137/0517059. [9] N. Laskin, Fractional Schrödinger equations, Phys. Rev. E, 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [10] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334. doi: 10.1016/0022-1236(82)90072-6. [11] C. Miranda, Un' osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7. [12] E. S. Noussair and J. Wei, On the effect of the domain geometry on the existence and profile of nodal solution of some singularly perturbed semilinear Dirichlet problem, Indiana Univ. Math. J., 46 (1997), 1255-1271. doi: 10.1512/iumj.1997.46.1401.
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