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Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum
Multiple solutions for superlinear elliptic systems of Hamiltonian type
1. | Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan |
2. | Department of Mathematics, Zhaotong Teacher’s College, Zhaotong 657000 Yunnan |
$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part,, Math. Z., 248 (2004), 423.
doi: 10.1007/s00209-004-0663-y. |
[2] |
N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations,, J. Func. Anal., 234 (2006), 277.
doi: 10.1016/j.jfa.2005.11.010. |
[3] |
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.
doi: 10.1016/S0022-247X(02)00413-4. |
[4] |
A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 459.
|
[5] |
A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, J. Differential Equations, 191 (2003), 348.
|
[6] |
T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems,, in, 35 (1999), 51.
|
[7] |
T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nach., 279 (2006), 1.
doi: 10.1002/mana.200410420. |
[8] |
V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals,, Inven. Math., 52 (1979), 241.
doi: 10.1007/BF01389883. |
[9] |
V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[10] |
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.
doi: 10.1002/cpa.3160451002. |
[11] |
D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Tran. Amer. Math. Soc., 355 (2003), 2973.
doi: 10.1090/S0002-9947-03-03257-4. |
[12] |
D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems,, Tran. Amer. Math. Soc., 343 (1994), 97.
|
[13] |
D. G. De Figueiredo, J. Marcos do Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems,, J. Func. Anal., 224 (2005), 471.
doi: 10.1016/j.jfa.2004.09.008. |
[14] |
D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211.
doi: 10.1016/S0362-546X(97)00548-8. |
[15] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdisciplinary Mathematical Sciences, 7 (2007).
doi: 10.1142/9789812709639. |
[16] |
Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.
doi: 10.1016/j.jde.2007.03.005. |
[17] |
Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.
|
[18] |
J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure,, J. Func. Anal., 114 (1993), 32.
doi: 10.1006/jfan.1993.1062. |
[19] |
W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications,, Tran. Amer. Math. Soc., 349 (1997), 3181.
doi: 10.1090/S0002-9947-97-01963-6. |
[20] |
W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441.
|
[21] |
G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, Comm. Contemp. Math., 4 (2002), 763.
doi: 10.1142/S0219199702000853. |
[22] |
G. Li and J. Yang, Asymptotically linear elliptic systems,, Comm. Partial Differential Equations, 29 (2004), 925.
|
[23] |
A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions,, J. Differential Equations, 201 (2004), 160.
|
[24] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978).
|
[25] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems,, Math. Z., 209 (1992), 133.
|
[26] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N,, Adv. Differential Equations, 5 (2000), 1445.
|
[27] |
C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation,, Comm. Partial Differential Equations, 21 (1996), 1431.
|
[28] |
J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems,, Nonlinear Anal., 72 (2010), 1949.
doi: 10.1016/j.na.2009.09.035. |
[29] | |
[30] |
J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N,, Electron. J. Diff. Eqns., 6 (2001), 343.
|
[31] |
F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems,, Nonlinear Differ. Equ. Appl., 15 (2008), 673.
|
[32] |
F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems,, ESAIM: Control, 16 (2010), 77.
doi: 10.1051/cocv:2008064. |
show all references
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part,, Math. Z., 248 (2004), 423.
doi: 10.1007/s00209-004-0663-y. |
[2] |
N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations,, J. Func. Anal., 234 (2006), 277.
doi: 10.1016/j.jfa.2005.11.010. |
[3] |
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.
doi: 10.1016/S0022-247X(02)00413-4. |
[4] |
A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 459.
|
[5] |
A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, J. Differential Equations, 191 (2003), 348.
|
[6] |
T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems,, in, 35 (1999), 51.
|
[7] |
T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nach., 279 (2006), 1.
doi: 10.1002/mana.200410420. |
[8] |
V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals,, Inven. Math., 52 (1979), 241.
doi: 10.1007/BF01389883. |
[9] |
V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.
doi: 10.1090/S0894-0347-1991-1119200-3. |
[10] |
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.
doi: 10.1002/cpa.3160451002. |
[11] |
D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Tran. Amer. Math. Soc., 355 (2003), 2973.
doi: 10.1090/S0002-9947-03-03257-4. |
[12] |
D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems,, Tran. Amer. Math. Soc., 343 (1994), 97.
|
[13] |
D. G. De Figueiredo, J. Marcos do Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems,, J. Func. Anal., 224 (2005), 471.
doi: 10.1016/j.jfa.2004.09.008. |
[14] |
D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211.
doi: 10.1016/S0362-546X(97)00548-8. |
[15] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdisciplinary Mathematical Sciences, 7 (2007).
doi: 10.1142/9789812709639. |
[16] |
Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.
doi: 10.1016/j.jde.2007.03.005. |
[17] |
Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.
|
[18] |
J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure,, J. Func. Anal., 114 (1993), 32.
doi: 10.1006/jfan.1993.1062. |
[19] |
W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications,, Tran. Amer. Math. Soc., 349 (1997), 3181.
doi: 10.1090/S0002-9947-97-01963-6. |
[20] |
W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441.
|
[21] |
G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, Comm. Contemp. Math., 4 (2002), 763.
doi: 10.1142/S0219199702000853. |
[22] |
G. Li and J. Yang, Asymptotically linear elliptic systems,, Comm. Partial Differential Equations, 29 (2004), 925.
|
[23] |
A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions,, J. Differential Equations, 201 (2004), 160.
|
[24] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978).
|
[25] |
E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems,, Math. Z., 209 (1992), 133.
|
[26] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N,, Adv. Differential Equations, 5 (2000), 1445.
|
[27] |
C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation,, Comm. Partial Differential Equations, 21 (1996), 1431.
|
[28] |
J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems,, Nonlinear Anal., 72 (2010), 1949.
doi: 10.1016/j.na.2009.09.035. |
[29] | |
[30] |
J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N,, Electron. J. Diff. Eqns., 6 (2001), 343.
|
[31] |
F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems,, Nonlinear Differ. Equ. Appl., 15 (2008), 673.
|
[32] |
F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems,, ESAIM: Control, 16 (2010), 77.
doi: 10.1051/cocv:2008064. |
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