MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER EQUATIONS WITH SATURABLE NONLINEARITY

doi:10.1007/s10473-021-0213-2

Abstract

Abstract: In this paper, we construct sign-changing radial solutions for a class of Schrödinger equations with saturable nonlinearity which arises from several models in mathematical physics. More precisely, for any given nonnegative integer $k$ , by using a minimization argument, we first obtain a sign-changing minimizer with $k$ nodes of a constrained minimization problem, and show, by a deformation lemma and Miranda's theorem, that the minimizer is the desired solution.

Key words: Sign-changing solutions, saturable nonlinearity, Nehari manifold, variational methods

CLC Number:

35J61

Zhongyuan LIU. MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER EQUATIONS WITH SATURABLE NONLINEARITY[J].Acta mathematica scientia,Series B, 2021, 41(2): 493-504.

Trendmd

References

[1] Akhmediev N, Ankiewicz A. Partially coherent solitons on a finite background. Phys Rev Lett, 1999, 82(13):2661-2664
[2] Agrawal G P, Kivshar Y S. Optical solitons:from fibers to photonic crystals. Academic Press, 2003
[3] Akhmediev N, Królinowski W, Snyder A. Partially coherent solitons of variable shape. Phys Rev Lett, 1998, 81(21):4632-4635
[4] Berestycki H, Lions P L. Nonlinear scalar field equations, I, existence of a ground state. Arch Ration Mech Anal, 1983, 82(4):313-345
[5] Berestycki H, Lions P L. Nonlinear scalar field equations, Ⅱ, existence of infinitely many solutions. Arch Ration Mech Anal, 1983, 82(4):347-375
[6] Bartsch T, Willem M. Infinitely many radial solutions of a semilinear elliptic problem on

$\mathbb{R}^N$ . Arch Ration Mech Anal, 1993, 124(3):261-276
[7] Castro A, Cossio J, Neuberger J M. A sigh-changing solution for a superlinear Dirichlet problem. Rocky Mountain J Math, 1997, 27(4):1041-1053
[8] Conti M, Merizzi L, Terracini S. Radial solutions of superlinear equations on

$R^N$ , Part I, A global variational approach. Arch Ration Mech Anal, 2000, 153(4):291-316
[9] Cao D, Li S, Liu Z. Nodal solutions for a supercritical semilinear problem with variable exponent. Cal Var PDEs, 2018, 57(2):38
[10] Cao D, Zhu X. On the existence and nodal character of solutions of semilinear elliptic equation. Acta Mathematica Scientia, 1988, 8B(3):285-300
[11] Cerami G, Solimini S, Struwe M. Some existence results for superlinear elliptic boundary problems involving critical exponents. J Funct Anal, 1986, 69(3):289-306
[12] Deng Y. The existence and nodal character of the solutions in

$\mathbb{R}^n$ for semilinear elliptic equation involving critical Sobolev exponent. Acta Mathematica Scientia, 1989, 9B(4):385-402
[13] Deng Y, Peng S, Shuai W. Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in

$\mathbb{R}^3$ . J Funct Anal, 2015, 269(11):3500-3527
[14] Deng Y, Peng S, Wang J. Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent. J Math Phys, 2013, 54(1):011504
[15] Deng Y, Peng S, Wang J. Nodal soliton solutions for generalized quasilinear Schrödinger equations. J Math Phys, 2014, 55(5):051501
[16] Kulpa W. The Poincaré-Miranda theorem. Amer Math Mon, 1997, 104(6):545-550
[17] Lin T C, Belić M R, Petrović M S, Chen G. Ground states of nonlinear Schrödinger systems with saturable nonlinearity in

$\mathbb{R}^2$ for two counterpropagating beams. J Math Phys, 2014, 55(1):011505
[18] Lin T C, Belić M R, Petrović M S, Aleksić N B, Chen G. Ground-state counterpropagating solitons in photorefractive media with saturable nonlinearity. J Opt Soc Am B, 2013, 30(4):1036-1040
[19] Lin T C, Belić M R, Petrović M S, Hajaiej H, Chen G. The virial theorem and ground state energy estimates of nonlinear Schrödinger equations in

$\mathbb{R}^2$ with square root and saturable nonlinearities in nonlinear optics. Cal Var PDEs, 2017, 56(5):147
[20] Liu T C, Wang X, Wang Z Q. Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in

$\mathbb{R}^2$ . Journal Diff Equations, 2017, 263(8):4750-4786
[21] Litchinitser N M, Królikowski W, Akhmediev N N, Agrawal G P. Asymmetric partially coherent solitons in saturable nonlinear media. Phys Rev E, 1999, 60(2):2377-2380
[22] Liu Z, Wang Z Q. On the Ambrosetti-Rabinowitz superlinear condition. Adv Nonlinear Stud, 2004, 4(4):563-574
[23] Maia L A, Miyagaki O H, Soares S. A sign-changing solution for an asymptotically linear Schrödinger equation. Proc Edin Math Soc, 2015, 58(3):697-716
[24] Miranda C. Un'osservazione su un teorema di Brouwer. Boll Un Mat Ital, 1940, 3(2):5-7
[25] Nehari Z. Characteristic values associated with a class of nonlinear second order differential equations. Acta Math, 1961, 105(3/4):141-175
[26] Ostrovskaya E A, Kivshar Y S. Multi-hump optical solitons in a saturable medium. J Opt B:Quantum Semiclassical Opt, 1999, 1(1):77-83
[27] Pohozaev S. Eigenfunctions of the equations △u +λf(u)=0. Dokl Akad Nauk SSSR, 1965, 165(1):36-39
[28] Ryder G H. Boundary value problem for a class of nonlinear differential equations. Pacific J Math, 1967, 22(3):477-503
[29] Struwe M. Superlinear elliptic boundary value problems with rotational symmetry. Arch Math, 1982, 39(3):233-240
[30] Stuart C A. Guidance properties of nonlinear planar waveguides. Arch Ration Mech Anal, 1993, 125(2):145-200
[31] Serrin J, Tang M. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ Math J, 2000, 49(3):897-923
[32] Stuart C A, Zhou H S. Applying the mountain pass theorem to an asymptotically linear elliptic equation on

$R^N$ . Commun PDEs, 1999, 24(9/10):1731-1758
[33] Stegeman G I, Christodoulides D N, Segev M. Optical spatial solitons:historical Perspectives. IEEE J Sel Top Quantum Electron, 2000, 6(6):1419-1427
[34] Szulkin A, Weth T. The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications. Boston:International Press, 2010:597-632
[35] Wang X, Liu T C, Wang Z Q. Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in

$\mathbb{R}^2$ . Nonlinear Anal, 2018, 173:19-36
[36] Willem M. Minimax Theorems. Basel:Birkhäser, 1996