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January  2018, 17(1): 143-161. doi: 10.3934/cpaa.2018009

Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

2. 

Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckich 8, 00-956 Warszawa, Poland

* Corresponding author

Received  February 2017 Revised  February 2017 Published  September 2017

Fund Project: The second author was supported by the National Science Centre, Poland (Grant No. 2014/15/D/ST1/03638).

We look for ground state solutions to the following nonlinear Schrödinger equation
$-Δ u + V(x)u = f(x,u)-Γ(x)|u|^{q-2}u\hbox{ on }\mathbb{R}^N,$
where $V=V_{per}+V_{loc}∈ L^{∞}(\mathbb{R}^N)$ is the sum of a periodic potential $V_{per}$ and a localized potential $V_{loc}$, $Γ∈ L^{∞}(\mathbb{R}^N)$ is periodic and $Γ(x)≥ 0$ for a.e. $x∈\mathbb{R}^N$ and $2≤q <2^*$. We assume that $\inf σ(-Δ+V)>0$, where $σ(-Δ+V)$ stands for the spectrum of $-Δ +V$ and $f$ has the subcritical growth but higher than $Γ(x)|u|^{q-2}u$, however the nonlinearity $f(x, u)-Γ(x)|u|^{q-2}u$ may change sign. Although a Nehari-type monotonicity condition for the nonlinearity is not satisfied, we investigate the existence of ground state solutions being minimizers on the Nehari manifold.
Citation: Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009
References:
[1]

N. Ackermann, Uniform continuity and Brézis-Lieb type splitting for superposition operators in Sobolev space, Advances in Nonlinear Analysis, (2016).  doi: 10.1515/anona-2016-0123.  Google Scholar

[2]

S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.  Google Scholar

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

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T. Bartsch and J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Rational Mech. Anal., 215 (2015), 283-306.  doi: 10.1007/s00205-014-0778-1.  Google Scholar

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T. Bartsch and J. Mederski, Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium, J. Funct. Anal., 272 (2017), 4304-4333.  doi: 10.1016/j.jfa.2017.02.019.  Google Scholar

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J. Belmonte-Beitia and D. Pelinovsky, Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient, Appl. Anal., 89 (2010), 1335-1350.  doi: 10.1080/00036810903330538.  Google Scholar

[7]

V. BenciC. R. Grisanti and A. M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with $V(∞) = 0$, Topol. Methods in Nonlinear Anal., 26 (2005), 203-219.  doi: 10.12775/TMNA.2005.031.  Google Scholar

[8] A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Chapman and Hall, 2006.   Google Scholar
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A. V. BuryakP. Di TrapaniD. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physic to futuristic applications, Physics Reports, 370 (2002), 63-235.  doi: 10.1016/S0370-1573(02)00196-5.  Google Scholar

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D. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbb{R}^N$, Cal. Var., 13, 159-189. doi: 10.1007/PL00009927.  Google Scholar

[11]

D. Costa and H. Tehrani, Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities, Adv. Differential Equations, 8 (2003), 1319-1340.   Google Scholar

[12]

V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.  doi: 10.1002/cpa.3160451002.  Google Scholar

[13] W. DörflerA. LechleiterM. PlumG. Schneider and C. Wieners, Photonic Crystals: Mathematical Analysis and Numerical Approximation, Springer, Basel, 2012.   Google Scholar
[14]

G. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795.  doi: 10.1512/iumj.2016.65.5828.  Google Scholar

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Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202.  doi: 10.1016/j.jde.2015.11.006.  Google Scholar

[16]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^N$, Indiana Univ. Math. Journal, 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.  Google Scholar

[17]

P. Kuchment, The mathematics of photonic crystals, Mathematical modeling in optical science, Frontiers Appl. Math. , 22, SIAM, Philadelphia (2001), 207-272.  Google Scholar

[18]

Y. LiZ.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[19]

P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré, Anal. Non Liné are., 1 (1984), 109-145; and 223-283.   Google Scholar

[20]

F. Liu and J. Yang, Nontrivial solutions of Schrödinger equations with indefinite nonlinearities, J. Math. Anal. Appl., 334 (2007), 627-645.  doi: 10.1016/j.jmaa.2006.12.054.  Google Scholar

[21]

S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

[22]

J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Methods Nonlinear Anal., 46 (2015), 755-771.  doi: 10.12775/TMNA.2015.067.  Google Scholar

[23]

J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations, 41 (2016), 1426-1440.  doi: 10.1080/03605302.2016.1209520.  Google Scholar

[24]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[25]

A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7.  Google Scholar

[26] A. Pankov, Lecture Notes on Schrödinger Equations, Nova Publ., 2007.   Google Scholar
[27]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[28]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978.  Google Scholar

[29]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.  Google Scholar

[30] R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals, Springer, 2003.   Google Scholar
[31] M. Struwe, Variational Methods, Springer, 2008.   Google Scholar
[32]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[33]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, 597-632, Int. Press, Somerville, 2010.  Google Scholar

[34]

X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Advanced Nonlinear Studies, 14 (2014), 361-373.  doi: 10.1515/ans-2014-0208.  Google Scholar

[35] M. Willem, Minimax Theorems, Birkhäuser Verlag, 1996.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

show all references

References:
[1]

N. Ackermann, Uniform continuity and Brézis-Lieb type splitting for superposition operators in Sobolev space, Advances in Nonlinear Analysis, (2016).  doi: 10.1515/anona-2016-0123.  Google Scholar

[2]

S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.  Google Scholar

[3]

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

[4]

T. Bartsch and J. Mederski, Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain, Arch. Rational Mech. Anal., 215 (2015), 283-306.  doi: 10.1007/s00205-014-0778-1.  Google Scholar

[5]

T. Bartsch and J. Mederski, Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium, J. Funct. Anal., 272 (2017), 4304-4333.  doi: 10.1016/j.jfa.2017.02.019.  Google Scholar

[6]

J. Belmonte-Beitia and D. Pelinovsky, Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient, Appl. Anal., 89 (2010), 1335-1350.  doi: 10.1080/00036810903330538.  Google Scholar

[7]

V. BenciC. R. Grisanti and A. M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with $V(∞) = 0$, Topol. Methods in Nonlinear Anal., 26 (2005), 203-219.  doi: 10.12775/TMNA.2005.031.  Google Scholar

[8] A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, Chapman and Hall, 2006.   Google Scholar
[9]

A. V. BuryakP. Di TrapaniD. V. Skryabin and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physic to futuristic applications, Physics Reports, 370 (2002), 63-235.  doi: 10.1016/S0370-1573(02)00196-5.  Google Scholar

[10]

D. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbb{R}^N$, Cal. Var., 13, 159-189. doi: 10.1007/PL00009927.  Google Scholar

[11]

D. Costa and H. Tehrani, Existence and multiplicity results for a class of Schrödinger equations with indefinite nonlinearities, Adv. Differential Equations, 8 (2003), 1319-1340.   Google Scholar

[12]

V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.  doi: 10.1002/cpa.3160451002.  Google Scholar

[13] W. DörflerA. LechleiterM. PlumG. Schneider and C. Wieners, Photonic Crystals: Mathematical Analysis and Numerical Approximation, Springer, Basel, 2012.   Google Scholar
[14]

G. Figueiredo and H. R. Quoirin, Ground states of elliptic problems involving non homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795.  doi: 10.1512/iumj.2016.65.5828.  Google Scholar

[15]

Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202.  doi: 10.1016/j.jde.2015.11.006.  Google Scholar

[16]

L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^N$, Indiana Univ. Math. Journal, 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.  Google Scholar

[17]

P. Kuchment, The mathematics of photonic crystals, Mathematical modeling in optical science, Frontiers Appl. Math. , 22, SIAM, Philadelphia (2001), 207-272.  Google Scholar

[18]

Y. LiZ.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837.  doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[19]

P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré, Anal. Non Liné are., 1 (1984), 109-145; and 223-283.   Google Scholar

[20]

F. Liu and J. Yang, Nontrivial solutions of Schrödinger equations with indefinite nonlinearities, J. Math. Anal. Appl., 334 (2007), 627-645.  doi: 10.1016/j.jmaa.2006.12.054.  Google Scholar

[21]

S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

[22]

J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Methods Nonlinear Anal., 46 (2015), 755-771.  doi: 10.12775/TMNA.2015.067.  Google Scholar

[23]

J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations, 41 (2016), 1426-1440.  doi: 10.1080/03605302.2016.1209520.  Google Scholar

[24]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[25]

A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7.  Google Scholar

[26] A. Pankov, Lecture Notes on Schrödinger Equations, Nova Publ., 2007.   Google Scholar
[27]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[28]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978.  Google Scholar

[29]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.  Google Scholar

[30] R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals, Springer, 2003.   Google Scholar
[31] M. Struwe, Variational Methods, Springer, 2008.   Google Scholar
[32]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[33]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, 597-632, Int. Press, Somerville, 2010.  Google Scholar

[34]

X. H. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Advanced Nonlinear Studies, 14 (2014), 361-373.  doi: 10.1515/ans-2014-0208.  Google Scholar

[35] M. Willem, Minimax Theorems, Birkhäuser Verlag, 1996.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar
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