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March  2022, 42(3): 1369-1401. doi: 10.3934/dcds.2021156

Elliptic systems involving Schrödinger operators with vanishing potentials

Juan Arratia 1, , Denilson Pereira 2, and  Pedro Ubilla 1,,

1. 

Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
2. 

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil

* Corresponding author: Pedro Ubilla

Received  February 2021 Revised  July 2021 Published  March 2022 Early access  November 2021

Fund Project: The second author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT. The third author was partially supported by FONDECYT Grant 1181125, 1161635, 1171691

We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators
$ \begin{equation*} \left\{ \begin{array}{cll} -\Delta u+V_{1}(x)u = \lambda\rho_{1}(x)(u+1)^{r}(v+1)^{p}&\mbox{ in }&\mathbb{R}^{N}\\ -\Delta v+V_{2}(x)v = \mu\rho_{2}(x)(u+1)^{q}(v+1)^{s}&\mbox{ in }&\mathbb{R}^{N},\\ u(x),v(x)\to 0& \mbox{ as}&|x|\to\infty \end{array} \right. \end{equation*} $
where
$ p,q,r,s\geq0 $
,
$ V_{i} $
 is a nonnegative vanishing potential, and
$ \rho_{i} $
 has the property
$ (\mathrm{H}) $
 introduced by Brezis and Kamin [4].As in that celebrated work we will prove that for every
$ R> 0 $
 there is a solution
$ (u_R, v_R) $
 defined on the ball of radius
$ R $
 centered at the origin. Then, we will show that this sequence of solutions tends to a bounded solution of the previous system when
$ R $
 tends to infinity. Furthermore, by imposing some restrictions on the powers
$ p,q,r,s $
 without additional hypotheses on the weights
$ \rho_{i} $
, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.
Keywords: Elliptic systems, gradient systems, Hamiltonian system, upper and lower solutions, variational methods, bounded solutions.
Mathematics Subject Classification: Primary: 35J20, 35J25, 35J60; Secondary: 47J30, 35B05.
Citation: Juan Arratia, Denilson Pereira, Pedro Ubilla. Elliptic systems involving Schrödinger operators with vanishing potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1369-1401. doi: 10.3934/dcds.2021156
References:
[1]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.

[2]

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

[3]

M.-F. Bidaut-Véron, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192. 

[4]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[5]

J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla, Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.  doi: 10.3934/dcds.2020392.

[6]

B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[8]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. 

[9]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[10]

G. Li and H. Ye, Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.  doi: 10.1016/S0252-9602(13)60050-8.

[11]

C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308.

[12]

E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32. 

[13]

M. Montenegro, The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229. 

[14]

P. Quittner, A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.  doi: 10.1016/j.na.2014.02.010.

[15]

P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.

[16]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258. 

[17]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.

[18]

C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285.

[19]

E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020.

[20]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.

[2]

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

[3]

M.-F. Bidaut-Véron, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192. 

[4]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[5]

J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla, Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.  doi: 10.3934/dcds.2020392.

[6]

B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[8]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. 

[9]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[10]

G. Li and H. Ye, Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.  doi: 10.1016/S0252-9602(13)60050-8.

[11]

C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308.

[12]

E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32. 

[13]

M. Montenegro, The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229. 

[14]

P. Quittner, A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.  doi: 10.1016/j.na.2014.02.010.

[15]

P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.

[16]

M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258. 

[17]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.

[18]

C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285.

[19]

E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020.

[20]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

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