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Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems
Elliptic systems involving Schrödinger operators with vanishing potentials
1. | Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile |
2. | Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil |
$ \begin{equation*} \left\{ \begin{array}{cll} -\Delta u+V_{1}(x)u = \lambda\rho_{1}(x)(u+1)^{r}(v+1)^{p}&\mbox{ in }&\mathbb{R}^{N}\\ -\Delta v+V_{2}(x)v = \mu\rho_{2}(x)(u+1)^{q}(v+1)^{s}&\mbox{ in }&\mathbb{R}^{N},\\ u(x),v(x)\to 0& \mbox{ as}&|x|\to\infty \end{array} \right. \end{equation*} $ |
$ p,q,r,s\geq0 $ |
$ V_{i} $ |
$ \rho_{i} $ |
$ (\mathrm{H}) $ |
$ R> 0 $ |
$ (u_R, v_R) $ |
$ R $ |
$ R $ |
$ p,q,r,s $ |
$ \rho_{i} $ |
References:
[1] |
A. Ambrosetti, V. Felli and A. Malchiodi,
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[2] |
H. Berestycki and P. L. Lions,
Nonlinear scalar fields equation I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[3] |
M.-F. Bidaut-Véron,
Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192.
|
[4] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[5] |
J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla,
Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.
doi: 10.3934/dcds.2020392. |
[6] |
B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[7] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[8] |
W. Kryszewski and A. Szulkin,
Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.
|
[9] |
G. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[10] |
G. Li and H. Ye,
Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.
doi: 10.1016/S0252-9602(13)60050-8. |
[11] |
C. R. Menyuk,
Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[12] |
E. Mitidieri and S. I. Pohozaev,
Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32.
|
[13] |
M. Montenegro,
The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229.
|
[14] |
P. Quittner,
A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.
doi: 10.1016/j.na.2014.02.010. |
[15] |
P. Quittner and P. Souplet,
Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.
doi: 10.1137/11085428X. |
[16] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[17] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[18] |
C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285. |
[19] |
E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020. |
[20] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
A. Ambrosetti, V. Felli and A. Malchiodi,
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[2] |
H. Berestycki and P. L. Lions,
Nonlinear scalar fields equation I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[3] |
M.-F. Bidaut-Véron,
Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192.
|
[4] |
H. Brezis and S. Kamin,
Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[5] |
J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla,
Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.
doi: 10.3934/dcds.2020392. |
[6] |
B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn,
Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594. |
[7] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001.
doi: 10.1007/978-3-642-61798-0. |
[8] |
W. Kryszewski and A. Szulkin,
Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.
|
[9] |
G. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[10] |
G. Li and H. Ye,
Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.
doi: 10.1016/S0252-9602(13)60050-8. |
[11] |
C. R. Menyuk,
Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.
doi: 10.1109/JQE.1987.1073308. |
[12] |
E. Mitidieri and S. I. Pohozaev,
Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32.
|
[13] |
M. Montenegro,
The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229.
|
[14] |
P. Quittner,
A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.
doi: 10.1016/j.na.2014.02.010. |
[15] |
P. Quittner and P. Souplet,
Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.
doi: 10.1137/11085428X. |
[16] |
M. A. S. Souto,
A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.
|
[17] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
doi: 10.1007/BF01626517. |
[18] |
C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285. |
[19] |
E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020. |
[20] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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