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Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity
On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms
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 |
$\left\{ \begin{array}{l}-\Delta u+V_{1}(x)u = f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}^{N},\\-\Delta v+V_{2}(x)v = f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right.$ |
$ N\geq 3 $ |
$ f_{1} $ |
$ f_{2} $ |
$ V_{1}(x) $ |
$ V_{2}(x) $ |
$ \lambda(x) $ |
$ \lambda(x)u $ |
$ \lambda(x)v $ |
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. |
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