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A nonlocal concave-convex problem with nonlocal mixed boundary data
Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth
1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400700, China |
2. | School of Mathematics and Statistics, Xin-Yang Normal University, Xinyang, 464000, China |
$-Δ u+V(x)u-Δ (u^2)u = K(x)|u|^{22^*-2}u+g(x,u), \ \ x∈ \mathbb{R}^N\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ |
References:
[1] |
S. Adachi and T. Watanabe,
Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833.
|
[2] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
|
[3] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.
|
[4] |
M. Colin, L. Jeanjean and M. Squassina,
Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385.
|
[5] |
Y. B. Deng, S. J. Peng and J. Wang,
Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.
|
[6] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262.
|
[7] |
J. M. do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.
|
[8] |
J.M. do Ó, O. H. Miyagaki and S. H. M. Soares,
Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744.
|
[9] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.
|
[10] |
F. Gladiali and M. Squassina,
Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179.
|
[11] |
Y. He and G. B. Li,
Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762.
|
[12] |
L. Jeanjean, T. J. Luo and Z. Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.
|
[13] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
|
[14] |
H. F. Lins and E. A. B. Silva,
Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.
|
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283.
|
[16] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145.
|
[17] |
J. Liu, J. F. Liao and C. L. Tang,
A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.
|
[18] |
J. Liu, J. F. Liao and C. L. Tang,
A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent, Comput. Math. Appl., 72 (2016), 1851-1864.
|
[19] |
J. Q. Liu, X. Q. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations, 39 (2014), 2216-2239.
|
[20] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
|
[21] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.
|
[22] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
|
[23] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
|
[24] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations with critical growth via perturbation method, Journal of Differential Equations, 254 (2013), 102-124.
|
[25] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669.
|
[26] |
R. D. Marchi,
Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757.
|
[27] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587.
|
[28] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
|
[29] |
D. Ruiz and G. Siciliano,
Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.
|
[30] |
A. Selvitella,
Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737.
|
[31] |
H. X. Shi and H. B. Chen,
Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.
|
[32] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.
|
[33] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.
|
[34] |
X. H. Tang,
Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.
|
[35] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
![]() |
[36] |
X. Wu,
Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.
|
[37] |
H. Zhang, J. X. Xu and F. B. Zhang,
On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.
|
[38] |
H. Zhang, J. X. Xu and F. B. Zhang,
Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122.
|
show all references
References:
[1] |
S. Adachi and T. Watanabe,
Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833.
|
[2] |
H. Brezis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
|
[3] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.
|
[4] |
M. Colin, L. Jeanjean and M. Squassina,
Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385.
|
[5] |
Y. B. Deng, S. J. Peng and J. Wang,
Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.
|
[6] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262.
|
[7] |
J. M. do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.
|
[8] |
J.M. do Ó, O. H. Miyagaki and S. H. M. Soares,
Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744.
|
[9] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.
|
[10] |
F. Gladiali and M. Squassina,
Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179.
|
[11] |
Y. He and G. B. Li,
Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762.
|
[12] |
L. Jeanjean, T. J. Luo and Z. Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.
|
[13] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
|
[14] |
H. F. Lins and E. A. B. Silva,
Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.
|
[15] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283.
|
[16] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145.
|
[17] |
J. Liu, J. F. Liao and C. L. Tang,
A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.
|
[18] |
J. Liu, J. F. Liao and C. L. Tang,
A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent, Comput. Math. Appl., 72 (2016), 1851-1864.
|
[19] |
J. Q. Liu, X. Q. Liu and Z. Q. Wang,
Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations, 39 (2014), 2216-2239.
|
[20] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.
|
[21] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.
|
[22] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
|
[23] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.
|
[24] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Quasilinear elliptic equations with critical growth via perturbation method, Journal of Differential Equations, 254 (2013), 102-124.
|
[25] |
X. Q. Liu, J. Q. Liu and Z. Q. Wang,
Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669.
|
[26] |
R. D. Marchi,
Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757.
|
[27] |
A. Moameni,
Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587.
|
[28] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.
|
[29] |
D. Ruiz and G. Siciliano,
Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.
|
[30] |
A. Selvitella,
Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737.
|
[31] |
H. X. Shi and H. B. Chen,
Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.
|
[32] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.
|
[33] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.
|
[34] |
X. H. Tang,
Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.
|
[35] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
![]() |
[36] |
X. Wu,
Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.
|
[37] |
H. Zhang, J. X. Xu and F. B. Zhang,
On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.
|
[38] |
H. Zhang, J. X. Xu and F. B. Zhang,
Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122.
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