On a class of linearly coupled systems on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> involving asymptotically linear terms

Edcarlos D. Silva; José Carlos de Albuquerque; Uberlandio Severo

doi:10.3934/cpaa.2019138

Previous Article
Molecular decomposition and a class of Fourier multipliers for bi-parameter modulation spaces
CPAA Home
This Issue
Next Article
Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	Z. Chen and W. Zou, On coupled systems of Schrödinger equations, Adv. Differential Equations, 16 (2011), 775-800.
[9]	D. G. Costa, On a class of elliptic systems in $\mathbb{R}^{N}$, Electron. J. Differential Equations, 7 (1994), 1-14.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[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.
[22]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
[23]	A. Szulkin and W. Andrzej, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, (2010), 597-632.
[24]	M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	Z. Chen and W. Zou, On coupled systems of Schrödinger equations, Adv. Differential Equations, 16 (2011), 775-800.
[9]	D. G. Costa, On a class of elliptic systems in $\mathbb{R}^{N}$, Electron. J. Differential Equations, 7 (1994), 1-14.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.
[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.
[22]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
[23]	A. Szulkin and W. Andrzej, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, (2010), 597-632.
[24]	M. Willem, Minimax Theorems, Birkhäser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[1]	Guanwei Chen, Martin Schechter. Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2107-2114. doi: 10.3934/dcdsb.2021124
[2]	Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[3]	Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations and Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069
[4]	Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105
[5]	Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
[6]	Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
[7]	Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial and Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
[8]	R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations and Control Theory, 2020, 9 (3) : 605-634. doi: 10.3934/eect.2020026
[9]	Yanxing Cui, Chuanlong Wang, Ruiping Wen. On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 863-873. doi: 10.3934/naco.2012.2.863
[10]	Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012
[11]	Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044
[12]	Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102
[13]	Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control and Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012
[14]	Yong-Fu Yang. Mechanism of the formation of singularities for diagonal systems with linearly degenerate characteristic fields. Communications on Pure and Applied Analysis, 2009, 8 (2) : 757-768. doi: 10.3934/cpaa.2009.8.757
[15]	Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128
[16]	P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213
[17]	Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127
[18]	Sara D. Cardell, Amparo Fúster-Sabater. Modelling the shrinking generator in terms of linear CA. Advances in Mathematics of Communications, 2016, 10 (4) : 797-809. doi: 10.3934/amc.2016041
[19]	Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[20]	Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors