April  2020, 40(4): 2165-2187. doi: 10.3934/dcds.2020110

Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions

1. 

Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan

2. 

Mathematics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan

3. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

*Corresponding author: T. F. Wu

Received  April 2019 Published  January 2020

Fund Project: T.C. Lin is partially supported by Center for Advanced Study in Theoretical Sciences (CASTS) and the Ministry of Science and Technology, Taiwan grant MOST-106-2115-M-002-003-MY3. T.F. Wu is partially supported by the Ministry of Science and Technology, Taiwan grant MOST-108-2115-M-390-007-MY2 and the National Center for Theoretical Sciences, Taiwan.

In this paper, we study saturable nonlinear Schrödinger equations with nonzero intensity function which makes the nonlinearity become not superlinear near zero. Using the Nehari manifold and the Lusternik-Schnirelman category, we prove the existence of multiple positive solutions for saturable nonlinear Schrödinger equations with nonzero intensity function which satisfies suitable conditions. The ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.

Citation: Tai-Chia Lin, Tsung-Fang Wu. Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110
References:
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Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Diff. Eqns., 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.  Google Scholar

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N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer and M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E, 66 (2002), 046602. doi: 10.1364/NLGW.2002.NLTuA4.  Google Scholar

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G. Li and H.-S. Zhou, The existence of a positive solution to asymptotically linear scalar field equation, Proc. R. Soc. Edinburgh A, 130 (2000), 81-105.  doi: 10.1017/S0308210500000068.  Google Scholar

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T.-C. Lin, M. R. Belic, M. S. Petrovic and G. Chen, Ground states of nonlinear Schrödinger systems with saturable nonlinearity in $\mathbb{R}^{2}$ for two counterpropagating beams, J. Math. Phys., 55 (2014), 011505, 13 pp. doi: 10.1063/1.4862190.  Google Scholar

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T.-C. Lin, M. R. Belic, M. S. Petrovic, H. Hajaiej and G. Chen, The virial theorem and ground state energy estimate of nonlinear Schrödinger equations in $\mathbb{R}^{2}$ with square root and saturable nonlinearities in nonlinear optics, Calc. Var. PDE, 56 (2017), Art. 147, 20 pp. doi: 10.1007/s00526-017-1251-4.  Google Scholar

[22]

T.-C. LinX. M. Wang and Z.-Q. Wang, Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in $\mathbb{R}^{2}$, J. Diff. Eqns., 263 (2017), 4750-4786.  doi: 10.1016/j.jde.2017.05.030.  Google Scholar

[23]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[24]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[25]

Z. L. Liu and Z.-Q. Wang, Existence of a positive solution of an elliptic equation on $\mathbb{R}^{N}$, Proc. R. Soc. Edinburgh A, 134 (2004), 191-200.  doi: 10.1017/S0308210500003152.  Google Scholar

[26]

C. Y. LiuZ. P. Wang and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Diff. Eqns., 245 (2008), 201-222.  doi: 10.1016/j.jde.2008.01.006.  Google Scholar

[27]

I. M. MerhasinB. A. MalomedK. SenthilnathanK. NakkeeranP. K. A. Wai and K. W. Chow, Solitons in Bragg gratings with saturable nonlinearities, J. Opt. Soc. Amer. B, 24 (2007), 1458-1468.  doi: 10.1364/JOSAB.24.001458.  Google Scholar

[28]

W.-M. Ni and I. Takagi, On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[29]

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[30]

C. A. Stuart and H. S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^{N}$, Commum. Partial Diff. Eqns., 24 (1999), 1731-1758.  doi: 10.1080/03605309908821481.  Google Scholar

[31]

H. Tehrani, A note on asymptotically linear elliptic problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 271 (2002), 546-554.  doi: 10.1016/S0022-247X(02)00143-9.  Google Scholar

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\triangle u+u = a(x)u^{p}+f(x)$ in $\mathbb{R}^{N}$, Calc. Var. PDE, 11 (2000), 63-95.  doi: 10.1007/s005260050003.  Google Scholar

[2]

A. Ambrosetti, Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), (1992), 139 pp.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[5]

K. J. Brown and Y. P. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Eqns., 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[6]

D. G. Costa and H. Tehrani, On a class of asymptotically linear elliptic problems in $\mathbb{R}^{N}$, J. Diff. Eqns., 173 (2001), 470-494.  doi: 10.1006/jdeq.2000.3944.  Google Scholar

[7]

Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Diff. Eqns., 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.  Google Scholar

[8]

A. L. Edelson and C. A. Stuart, The principle branch of solutions of a nonlinear elliptic eigenvalue problem on $\mathbb{R}^{N}$, J. Diff. Eqns., 124 (1996), 279-301.  doi: 10.1006/jdeq.1996.0010.  Google Scholar

[9]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer and M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E, 66 (2002), 046602. doi: 10.1364/NLGW.2002.NLTuA4.  Google Scholar

[10]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen and M. Segev, Two-dimensional optical lattice solitons, Phys. Rev. Lett., 91 (2003), 213906. doi: 10.1103/PhysRevLett.91.213906.  Google Scholar

[11]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[12]

S. Gatz and J. Herrmann, Propagation of optical beams and the properties of two-dimensional spatial solitons in media with a local saturable nonlinear refractive index, J. Opt. Soc. Amer. B, 14 (1997), 1795-1806.  doi: 10.1364/JOSAB.14.001795.  Google Scholar

[13]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1978), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[14]

L. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesmann-Lazer type problem, Proc. R. Soc. Edinburgh A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[15]

L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^{N}$ autinomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614.  doi: 10.1051/cocv:2002068.  Google Scholar

[16]

L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. PDE, 21 (2004), 287-318.  doi: 10.1007/s00526-003-0261-6.  Google Scholar

[17]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^{N}$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[18]

D. JovicR. JovanovicS. PrvanovicM. Petrovic and M. Belic, Counterpropagating beams in rotationally symmetric photonic lattices, Opt. Mater., 30 (2008), 1173-1176.   Google Scholar

[19]

G. Li and H.-S. Zhou, The existence of a positive solution to asymptotically linear scalar field equation, Proc. R. Soc. Edinburgh A, 130 (2000), 81-105.  doi: 10.1017/S0308210500000068.  Google Scholar

[20]

T.-C. Lin, M. R. Belic, M. S. Petrovic and G. Chen, Ground states of nonlinear Schrödinger systems with saturable nonlinearity in $\mathbb{R}^{2}$ for two counterpropagating beams, J. Math. Phys., 55 (2014), 011505, 13 pp. doi: 10.1063/1.4862190.  Google Scholar

[21]

T.-C. Lin, M. R. Belic, M. S. Petrovic, H. Hajaiej and G. Chen, The virial theorem and ground state energy estimate of nonlinear Schrödinger equations in $\mathbb{R}^{2}$ with square root and saturable nonlinearities in nonlinear optics, Calc. Var. PDE, 56 (2017), Art. 147, 20 pp. doi: 10.1007/s00526-017-1251-4.  Google Scholar

[22]

T.-C. LinX. M. Wang and Z.-Q. Wang, Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in $\mathbb{R}^{2}$, J. Diff. Eqns., 263 (2017), 4750-4786.  doi: 10.1016/j.jde.2017.05.030.  Google Scholar

[23]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[24]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[25]

Z. L. Liu and Z.-Q. Wang, Existence of a positive solution of an elliptic equation on $\mathbb{R}^{N}$, Proc. R. Soc. Edinburgh A, 134 (2004), 191-200.  doi: 10.1017/S0308210500003152.  Google Scholar

[26]

C. Y. LiuZ. P. Wang and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Diff. Eqns., 245 (2008), 201-222.  doi: 10.1016/j.jde.2008.01.006.  Google Scholar

[27]

I. M. MerhasinB. A. MalomedK. SenthilnathanK. NakkeeranP. K. A. Wai and K. W. Chow, Solitons in Bragg gratings with saturable nonlinearities, J. Opt. Soc. Amer. B, 24 (2007), 1458-1468.  doi: 10.1364/JOSAB.24.001458.  Google Scholar

[28]

W.-M. Ni and I. Takagi, On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[29]

J. Serrin and M. X. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana University Mathematics Journal, 49 (2000), 897-923.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar

[30]

C. A. Stuart and H. S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^{N}$, Commum. Partial Diff. Eqns., 24 (1999), 1731-1758.  doi: 10.1080/03605309908821481.  Google Scholar

[31]

H. Tehrani, A note on asymptotically linear elliptic problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 271 (2002), 546-554.  doi: 10.1016/S0022-247X(02)00143-9.  Google Scholar

[32]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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