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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 |
$ -\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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954. |
[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. |
[17] |
J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972. |
[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. |
[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. |
[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. |
[21] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978. |
[22] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160.
doi: 10.1007/BF02570817. |
[23] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464. |
[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. |
[25] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978. |
[26] |
J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273.
doi: 10.1007/BF01389883. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954. |
[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. |
[17] |
J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972. |
[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. |
[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. |
[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. |
[21] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978. |
[22] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160.
doi: 10.1007/BF02570817. |
[23] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464. |
[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. |
[25] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978. |
[26] |
J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357. |
[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. |
[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. |
[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. |
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