# American Institute of Mathematical Sciences

July  2011, 10(4): 1149-1163. doi: 10.3934/cpaa.2011.10.1149

## A note on a superlinear and periodic elliptic system in the whole space

 1 Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnanage, China 2 Office of Adult Education, Simao Teacher's College, Simao 665000 Yunnan, China 3 Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan

Received  April 2010 Revised  December 2010 Published  April 2011

This paper is concerned with the following periodic Hamiltonian elliptic system

$-\Delta u+V(x)u=g(x,v)$ in $R^N,$

$-\Delta v+V(x)v=f(x,u)$ in $R^N,$

$u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$

where the potential $V$ is periodic and has a positive bound from below, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear but subcritical in $t$ at infinity. By using generalized Nehari manifold method, existence of a positive ground state solution as well as multiple solutions for odd $f$ and $g$ are obtained.

Citation: Shuying He, Rumei Zhang, Fukun Zhao. A note on a superlinear and periodic elliptic system in the whole space. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1149-1163. doi: 10.3934/cpaa.2011.10.1149
##### References:
 [1] C. O. Alves, P. C. Carrião and O. H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in $\mathbbR^N$, J. Math. Anal. Appl., 276 (2002), 673-690. doi: 10.1016/S0022-247X(02)00413-4.  Google Scholar [2] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Diff. Eqns. Appl., 12 (2005), 459-479. doi: 10.1007/s00030-005-0022-7.  Google Scholar [3] A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Diff. Eqns., 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2.  Google Scholar [4] T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 35, Birkhäuser, Basel/Switzerland, 1999, 51-67.  Google Scholar [5] T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1-22. doi: 10.1002/mana.200410420.  Google Scholar [6] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.  Google Scholar [7] V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar [8] D. G. De Figueiredo and Y. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems, Tran. Amer. Math. Soc., 355 (2003), 2973-2989. doi: 10.1090/S0002-9947-03-03257-4.  Google Scholar [9] D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Tran. Amer. Math. Soc., 343 (1994), 97-116. doi: 10.1090/S0002-9947-1994-1214781-2.  Google Scholar [10] D. G. De Figueiredo, J. M. DO Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems, J. Func. Anal., 224 (2005), 471-496. doi: 10.1016/j.jfa.2004.09.008.  Google Scholar [11] D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar [12] J. Hulshof and R. C. A. M. Van der Vorst, Differential systems with strongly variational structure, J. Func. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.  Google Scholar [13] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.  Google Scholar [14] W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Tran. Amer. Math. Soc., 349 (1997), 3181-3234. doi: 10.1090/S0002-9947-97-01963-6.  Google Scholar [15] G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.  Google Scholar [16] Y. Li, Z. 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 [17] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972.  Google Scholar [18] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré, Analyse non linéaire, 1 (1984), 223-283.  Google Scholar [19] A. Pankov, Periodic nonlinear Schröinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  Google Scholar [20] A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions, J. Diff. Eqns., 201 (2004), 160-176. doi: 10.1016/j.jde.2004.02.003.  Google Scholar [21] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.  Google Scholar [22] E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.  Google Scholar [23] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.  Google Scholar [24] A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar [25] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar [26] J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.  Google Scholar [27] F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688. doi: 10.1007/s00030-008-7080-6.  Google Scholar [28] F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 77-91. doi: 10.1051/cocv:2008064.  Google Scholar [29] F. Zhao, L. Zhao and Y. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120.  Google Scholar

show all references

##### References:
 [1] C. O. Alves, P. C. Carrião and O. H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in $\mathbbR^N$, J. Math. Anal. Appl., 276 (2002), 673-690. doi: 10.1016/S0022-247X(02)00413-4.  Google Scholar [2] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Diff. Eqns. Appl., 12 (2005), 459-479. doi: 10.1007/s00030-005-0022-7.  Google Scholar [3] A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Diff. Eqns., 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2.  Google Scholar [4] T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 35, Birkhäuser, Basel/Switzerland, 1999, 51-67.  Google Scholar [5] T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1-22. doi: 10.1002/mana.200410420.  Google Scholar [6] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.  Google Scholar [7] V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.  Google Scholar [8] D. G. De Figueiredo and Y. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems, Tran. Amer. Math. Soc., 355 (2003), 2973-2989. doi: 10.1090/S0002-9947-03-03257-4.  Google Scholar [9] D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Tran. Amer. Math. Soc., 343 (1994), 97-116. doi: 10.1090/S0002-9947-1994-1214781-2.  Google Scholar [10] D. G. De Figueiredo, J. M. DO Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems, J. Func. Anal., 224 (2005), 471-496. doi: 10.1016/j.jfa.2004.09.008.  Google Scholar [11] D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar [12] J. Hulshof and R. C. A. M. Van der Vorst, Differential systems with strongly variational structure, J. Func. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.  Google Scholar [13] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.  Google Scholar [14] W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Tran. Amer. Math. Soc., 349 (1997), 3181-3234. doi: 10.1090/S0002-9947-97-01963-6.  Google Scholar [15] G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.  Google Scholar [16] Y. Li, Z. 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 [17] J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972.  Google Scholar [18] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré, Analyse non linéaire, 1 (1984), 223-283.  Google Scholar [19] A. Pankov, Periodic nonlinear Schröinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  Google Scholar [20] A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions, J. Diff. Eqns., 201 (2004), 160-176. doi: 10.1016/j.jde.2004.02.003.  Google Scholar [21] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.  Google Scholar [22] E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.  Google Scholar [23] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.  Google Scholar [24] A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar [25] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar [26] J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.  Google Scholar [27] F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688. doi: 10.1007/s00030-008-7080-6.  Google Scholar [28] F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 77-91. doi: 10.1051/cocv:2008064.  Google Scholar [29] F. Zhao, L. Zhao and Y. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120.  Google Scholar
 [1] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107 [2] Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599 [3] Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335 [4] Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333 [5] Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151 [6] Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195 [7] Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110 [8] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. Homogenization of variational functionals with nonstandard growth in perforated domains. Networks & Heterogeneous Media, 2010, 5 (2) : 189-215. doi: 10.3934/nhm.2010.5.189 [9] Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435 [10] Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383 [11] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929 [12] B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 [13] Henrik Garde, Stratos Staboulis. The regularized monotonicity method: Detecting irregular indefinite inclusions. Inverse Problems & Imaging, 2019, 13 (1) : 93-116. doi: 10.3934/ipi.2019006 [14] Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126 [15] Paula Balseiro, Teresinha J. Stuchi, Alejandro Cabrera, Jair Koiller. About simple variational splines from the Hamiltonian viewpoint. Journal of Geometric Mechanics, 2017, 9 (3) : 257-290. doi: 10.3934/jgm.2017011 [16] Sandro Zagatti. Minimization of non quasiconvex functionals by integro-extremization method. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 625-641. doi: 10.3934/dcds.2008.21.625 [17] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 [18] Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315 [19] M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703 [20] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277

2019 Impact Factor: 1.105