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November  2019, 18(6): 3089-3101. doi: 10.3934/cpaa.2019138

On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms

Edcarlos D. Silva 1,, , José Carlos de Albuquerque 2, and  Uberlandio Severo 3,

1. 

Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970, Goiás–GO, Brazil
2. 

Departamento de Matemática, Universidade Federal de Pernambuco, 50670-901, Recife–PE, Brazil
3. 

Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa–PB, Brazil

* Corresponding author

Received  October 2018 Revised  January 2019 Published  May 2019

Fund Project: Research supported by CNPq and CAPES. The first author was also partially supported by Fapeg/CNPq grants 03/2015-PPP.

In this work we study the existence of positive solutions for the following class of coupled elliptic systems involving nonlinear Schrödinger equations
$\left\{ \begin{array}{l}-\Delta u+V_{1}(x)u = f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}^{N},\\-\Delta v+V_{2}(x)v = f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right.$
where
$ N\geq 3 $
 and the nonlinearities
$ f_{1} $
 and
$ f_{2} $
 are asymptotically linear at infinity. The potentials
$ V_{1}(x) $
 and
$ V_{2}(x) $
 are continuous functions which are bounded from below and above. The function
$ \lambda(x) $
 is continuous and gives us a linear coupling due the terms
$ \lambda(x)u $
 and
$ \lambda(x)v $
. Here we employ some variational arguments jointly with a Pohozaev identity.
Keywords: Linearly coupled systems, asymptotically linear terms, variational methods.
Mathematics Subject Classification: Primary: 35J47, 35J50; Secondary: 35Q55.
Citation: Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138
References:
[1]

A. Ambrosetti, Remarks on some systems of nonlinear Schrödinger equations, Fixed Point Theory Appl., 4 (2008), 35-46.  doi: 10.1007/s11784-007-0035-4.  Google Scholar

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

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A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R}^N$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013.  Google Scholar

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[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅱ. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[7]

Z. Chen and W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107.  doi: 10.1016/j.jfa.2012.01.001.  Google Scholar

[8]

Z. Chen and W. Zou, On coupled systems of Schrödinger equations, Adv. Differential Equations, 16 (2011), 775-800.   Google Scholar

[9]

D. G. Costa, On a class of elliptic systems in $\mathbb{R}^{N}$, Electron. J. Differential Equations, 7 (1994), 1-14.   Google Scholar

[10]

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

[11]

M. F. Furtado, E. A. B. Silva and M. S. Xavier, Multiplicity and concentration of solutions for elliptic systems with vanishing potentials, J. Differential Equations, 249 (2010), 2377-2396.  doi: 10.1016/j.jde.2010.08.002.  Google Scholar

[12]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Solutions for a resonant elliptic system with coupling in $ \mathbb{R}^N $, Comm. Partial Differential Equations, 27 (2002), 1515-1536.  doi: 10.1081/PDE-120005847.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer-type set on $ \mathbb{R}^{N} $, Proc. Roy. Soc. Edinburgh Sect. 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} $ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614.  doi: 10.1051/cocv:2002068.  Google Scholar

[16]

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

[17]

L. Lions and K. Tanaka, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

[18]

L. A. Maia and E. A. B. Silva, On a class of coupled elliptic systems in $ \mathbb{R}^{N} $, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 303-313.  doi: 10.1007/s00030-007-5039-7.  Google Scholar

[19]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[20]

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

[21]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491-502.  doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[22]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[23]

A. Szulkin and W. Andrzej, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, (2010), 597-632.   Google Scholar

[24]

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

show all references

References:
[1]

A. Ambrosetti, Remarks on some systems of nonlinear Schrödinger equations, Fixed Point Theory Appl., 4 (2008), 35-46.  doi: 10.1007/s11784-007-0035-4.  Google Scholar

[2]

A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024.  Google Scholar

[3]

A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb{R}^N$, J. Funct. Anal., 254 (2008), 2816-2845.  doi: 10.1016/j.jfa.2007.11.013.  Google Scholar

[4]

N. Akhmediev and A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett., 70 (1993), 2395-2398.  doi: 10.1103/PhysRevLett.70.2395.  Google Scholar

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅱ. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar

[7]

Z. Chen and W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal., 262 (2012), 3091-3107.  doi: 10.1016/j.jfa.2012.01.001.  Google Scholar

[8]

Z. Chen and W. Zou, On coupled systems of Schrödinger equations, Adv. Differential Equations, 16 (2011), 775-800.   Google Scholar

[9]

D. G. Costa, On a class of elliptic systems in $\mathbb{R}^{N}$, Electron. J. Differential Equations, 7 (1994), 1-14.   Google Scholar

[10]

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

[11]

M. F. Furtado, E. A. B. Silva and M. S. Xavier, Multiplicity and concentration of solutions for elliptic systems with vanishing potentials, J. Differential Equations, 249 (2010), 2377-2396.  doi: 10.1016/j.jde.2010.08.002.  Google Scholar

[12]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Solutions for a resonant elliptic system with coupling in $ \mathbb{R}^N $, Comm. Partial Differential Equations, 27 (2002), 1515-1536.  doi: 10.1081/PDE-120005847.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer-type set on $ \mathbb{R}^{N} $, Proc. Roy. Soc. Edinburgh Sect. 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} $ autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597-614.  doi: 10.1051/cocv:2002068.  Google Scholar

[16]

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

[17]

L. Lions and K. Tanaka, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.   Google Scholar

[18]

L. A. Maia and E. A. B. Silva, On a class of coupled elliptic systems in $ \mathbb{R}^{N} $, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 303-313.  doi: 10.1007/s00030-007-5039-7.  Google Scholar

[19]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.  Google Scholar

[20]

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

[21]

M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491-502.  doi: 10.1112/jlms/s2-44.3.491.  Google Scholar

[22]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[23]

A. Szulkin and W. Andrzej, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, (2010), 597-632.   Google Scholar

[24]

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

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