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 and Continuous Dynamical Systems, 2020, 40 (4) : 2165-2187. doi: 10.3934/dcds.2020110
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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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