# American Institute of Mathematical Sciences

February  2016, 36(2): 917-939. doi: 10.3934/dcds.2016.36.917

## Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China 2 Department of Mathematics, Huazhong Normal University,Wuhan, 430079

Received  March 2014 Revised  January 2015 Published  August 2015

We consider the following nonlinear fractional scalar field equation $$(-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N,$$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
Citation: Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
##### References:
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Anal., 258 (2010), 3048.  doi: 10.1016/j.jfa.2009.12.008.  Google Scholar [40] M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Comm. Partial Differential Equations, 12 (1987), 1133.  doi: 10.1080/03605308708820522.  Google Scholar

show all references

##### References:
 [1] L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves,, Phys. D, 40 (1989), 360.  doi: 10.1016/0167-2789(89)90050-X.  Google Scholar [2] W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials,, Calc. Var. Partial Differential Equations, 51 (2014), 761.  doi: 10.1007/s00526-013-0694-5.  Google Scholar [3] J. Byeon and Y. Oshita, Existence of multi-bump stading waves with a critical frequency for nonlinear schrödinger equations,, Comm. Partial Differential Equations, 29 (2005), 1877.  doi: 10.1081/PDE-200040205.  Google Scholar [4] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar [5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245.  doi: 10.1080/03605300600987306.  Google Scholar [6] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, Math. Ann., 336 (2006), 925.  doi: 10.1007/s00208-006-0021-y.  Google Scholar [7] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations,, Comm. Partial Differential Equations, 36 (2011), 1353.  doi: 10.1080/03605302.2011.562954.  Google Scholar [8] G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations,, Calc. Var. Partial Differential Equations, 23 (2005), 139.  doi: 10.1007/s00526-004-0293-6.  Google Scholar [9] G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equation with non-symmetric coefficients,, Comm. Pure Appl. Math., 66 (2013), 372.  doi: 10.1002/cpa.21410.  Google Scholar [10] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves,, SIAM. J. Math. Anal., 39 (): 1070.  doi: 10.1137/050648389.  Google Scholar [11] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar [12] G. Chen and Y. Zhang, Concentration phenomenon for fractionsl nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359.  doi: 10.3934/cpaa.2014.13.2359.  Google Scholar [13] T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423.  doi: 10.1016/j.anihpc.2009.01.002.  Google Scholar [14] J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for fractional nonlinear Schrödinger equation,, J. Differerntial Equations, 256 (2014), 858.  doi: 10.1016/j.jde.2013.10.006.  Google Scholar [15] M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains,, Calc. Var. Partial Differential Equations, 4 (1996), 121.  doi: 10.1007/BF01189950.  Google Scholar [16] G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations,, Adv. Nonlinear Stud., 12 (2012), 173.   Google Scholar [17] W. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, Arch. Ration. Mech. Anal., 91 (1986), 283.  doi: 10.1007/BF00282336.  Google Scholar [18] P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237.  doi: 10.1017/S0308210511000746.  Google Scholar [19] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbbR$,, Acta Math., 210 (2013), 261.  doi: 10.1007/s11511-013-0095-9.  Google Scholar [20] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , ().   Google Scholar [21] A. Elgart and B. Schlein, Mean field dynamics of boson stars,, Comm. Pure Appl. Math., 60 (2007), 500.  doi: 10.1002/cpa.20134.  Google Scholar [22] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899.   Google Scholar [23] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbbR^n$,, Arch. Ration. Mech. Anal., 105 (1989), 243.  doi: 10.1007/BF00251502.  Google Scholar [24] N. Laskin, Fractional quantum mechanics and Lévy path integrals,, Phys. Lett. A, 268 (2000), 298.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar [25] N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002).  doi: 10.1103/PhysRevE.66.056108.  Google Scholar [26] A. J. Majda, D. W. McLaughlin and E. G. Tabak, A one-dimensional model for dispersive wave turbulence,, J. Nonlinear Sci., 7 (1997), 9.  doi: 10.1007/BF02679124.  Google Scholar [27] M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation,, Nonlinear Anal., 51 (2002), 1073.  doi: 10.1016/S0362-546X(01)00880-X.  Google Scholar [28] E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem,, J. Lond. Math. Soc., 62 (2000), 213.  doi: 10.1112/S002461070000898X.  Google Scholar [29] E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems,, Proc. London Math. Soc., 76 (1998), 427.  doi: 10.1112/S0024611598000148.  Google Scholar [30] E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics,, Comm. Math. Phys., 112 (1987), 147.  doi: 10.1007/BF01217684.  Google Scholar [31] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109.   Google Scholar [32] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223.   Google Scholar [33] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar [34] G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces,, Calc. Var. Partial Differential Equations, 50 (2014), 799.  doi: 10.1007/s00526-013-0656-y.  Google Scholar [35] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, J. Funct. Anal., 256 (2009), 1842.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar [36] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21.  doi: 10.1007/s00526-010-0378-3.  Google Scholar [37] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229.  doi: 10.1007/BF02096642.  Google Scholar [38] J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in $\mathbbR^n$,, Calc. Var. Partial Differential Equations, 37 (2010), 423.  doi: 10.1007/s00526-009-0270-1.  Google Scholar [39] J. Wei and S. Yan, Infinite many positive solutions for the prescribed scalar curvature problem on $\mathbbS^N$,, J. Funct. Anal., 258 (2010), 3048.  doi: 10.1016/j.jfa.2009.12.008.  Google Scholar [40] M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, Comm. Partial Differential Equations, 12 (1987), 1133.  doi: 10.1080/03605308708820522.  Google Scholar
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