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A positive solution for an asymptotically cubic quasilinear Schrödinger equation
School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China |
$ - \Delta u + V(x)u - \Delta ({u^2})u = q(x)g(u),\;\;\;\;x \in {\mathbb{R}^N}, $ |
$N≥ 1$ |
$0 < q(x)≤ \lim_{|x|\to∞}q(x)$ |
$g∈ C(\mathbb{R}^+, \mathbb{R})$ |
$g(u)/u^3 \to 1$ |
$u \to ∞.$ |
References:
[1] |
S. Adachi and T. Watanabe,
Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal., 75 (2012), 819-833.
doi: 10.1016/j.na.2011.09.015. |
[2] |
S. Adachi, M. Shibata and T. Watanabe,
Global uniqueness results for ground states for a class of quasilinear elliptic equations, Kodai Math. J., 40 (2017), 117-142.
doi: 10.2996/kmj/1490083227. |
[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] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
P. C. Carrião, R. Lehrer and O. H. Miyagaki,
Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb{R}^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.
doi: 10.1016/j.jmaa.2015.02.060. |
[6] |
G. Cerami and D. Passaseo,
The effect of concentrating potentials in some singularly perturbed problems, Calc. Var., 17 (2003), 257-281.
|
[7] |
K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.
![]() ![]() |
[8] |
M. Clapp and L. A. Maia,
A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Diff. Eq., 260 (2016), 3173-3192.
doi: 10.1016/j.jde.2015.09.059. |
[9] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: a dual approach, Nonl. Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[10] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in $\mathbb{R}^N$, J. Diff. Eq., 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[11] |
J. M. do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure Appl. Anal., 9 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[12] |
J. M. do Ó and U. Severo,
Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var., 38 (2010), 275-315.
doi: 10.1007/s00526-009-0286-6. |
[13] |
G. Evéquoz and T. Weth,
Entire solutions to nonlinear scalar field equations with indefinite linear part, Adv. Nonlinear Stud., 12 (2012), 281-314.
doi: 10.1515/ans-2012-0206. |
[14] |
X. D. Fang and Z. Q. Han,
Existence of a Ground State Solution for a Quasilinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 941-950.
doi: 10.1515/ans-2014-0407. |
[15] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Diff. Eq., 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[16] |
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. |
[17] |
R. Lehrer and L. A. Maia,
Positive solutions of asymptotically linear equations via Pohozaev manifold, J. Funct. Anal., 266 (2014), 213-246.
doi: 10.1016/j.jfa.2013.09.002. |
[18] |
R. Lehrer, L. A. Maia and R. Ruviaro,
Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential, Nonlinear Diff. Equ. Appl., 22 (2015), 651-672.
doi: 10.1007/s00030-014-0299-5. |
[19] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Diff. Eq., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[21] |
X. Q. Liu, Y. S. Huang and J. Q. Liu,
Sign-changing solutions for an asymptotically linear Schrödinger equation with deepening potential well, Adv. Diff. Eq., 16 (2011), 1-30.
|
[22] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[23] |
A. Selvitella,
Nondegeneracy of the ground state for quasilinear Schrödinger equations, Calc. Var., 53 (2015), 349-364.
doi: 10.1007/s00526-014-0751-8. |
[24] |
M. Struwe, Variational Methods, second ed., Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03212-1.![]() ![]() ![]() |
[25] |
C. A. Stuart, An introduction to elliptic equation on $\mathbb{R}^N$, in Nonlinear Functional Analysis and Applications to Differential Equations (A. Ambrosetti, K.-C. Chang and I. Ekeland eds.), World Scientific, Singapore, 1998. |
[26] |
C. A. Stuart and H. S. Zhou,
Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^N$, Comm. Partial Diff. Eq., 24 (1999), 1731-1758.
doi: 10.1080/03605309908821481. |
[27] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[28] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis
and Applications, Int. Press, (2010), 597-632. |
[29] |
M. Willem, Minimax Theorems, in Progress in Nonlinear Differential Equations and Their
Applications, 24 , Birkhäuser Boston, Inc., Boston, (1996), ⅹ-162.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
Y. J. Wang and W. M. Zou,
Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Eq. Appl., 19 (2012), 19-47.
doi: 10.1007/s00030-011-0116-3. |
[31] |
M. B. Yang and Y. H. Ding,
Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in $\mathbb{R}^N$, Ann. Mat. Pura Appl., 192 (2013), 783-804.
doi: 10.1007/s10231-011-0246-6. |
show all references
References:
[1] |
S. Adachi and T. Watanabe,
Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal., 75 (2012), 819-833.
doi: 10.1016/j.na.2011.09.015. |
[2] |
S. Adachi, M. Shibata and T. Watanabe,
Global uniqueness results for ground states for a class of quasilinear elliptic equations, Kodai Math. J., 40 (2017), 117-142.
doi: 10.2996/kmj/1490083227. |
[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] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations, Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[5] |
P. C. Carrião, R. Lehrer and O. H. Miyagaki,
Existence of solutions to a class of asymptotically linear Schrödinger equations in $\mathbb{R}^N$ via the Pohozaev manifold, J. Math. Anal. Appl., 428 (2015), 165-183.
doi: 10.1016/j.jmaa.2015.02.060. |
[6] |
G. Cerami and D. Passaseo,
The effect of concentrating potentials in some singularly perturbed problems, Calc. Var., 17 (2003), 257-281.
|
[7] |
K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.
![]() ![]() |
[8] |
M. Clapp and L. A. Maia,
A positive bound state for an asymptotically linear or superlinear Schrödinger equation, J. Diff. Eq., 260 (2016), 3173-3192.
doi: 10.1016/j.jde.2015.09.059. |
[9] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equation: a dual approach, Nonl. Anal., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[10] |
D. G. Costa and H. Tehrani,
On a class of asymptotically linear elliptic problems in $\mathbb{R}^N$, J. Diff. Eq., 173 (2001), 470-494.
doi: 10.1006/jdeq.2000.3944. |
[11] |
J. M. do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure Appl. Anal., 9 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[12] |
J. M. do Ó and U. Severo,
Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var., 38 (2010), 275-315.
doi: 10.1007/s00526-009-0286-6. |
[13] |
G. Evéquoz and T. Weth,
Entire solutions to nonlinear scalar field equations with indefinite linear part, Adv. Nonlinear Stud., 12 (2012), 281-314.
doi: 10.1515/ans-2012-0206. |
[14] |
X. D. Fang and Z. Q. Han,
Existence of a Ground State Solution for a Quasilinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 941-950.
doi: 10.1515/ans-2014-0407. |
[15] |
X. D. Fang and A. Szulkin,
Multiple solutions for a quasilinear Schrödinger equation, J. Diff. Eq., 254 (2013), 2015-2032.
doi: 10.1016/j.jde.2012.11.017. |
[16] |
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. |
[17] |
R. Lehrer and L. A. Maia,
Positive solutions of asymptotically linear equations via Pohozaev manifold, J. Funct. Anal., 266 (2014), 213-246.
doi: 10.1016/j.jfa.2013.09.002. |
[18] |
R. Lehrer, L. A. Maia and R. Ruviaro,
Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential, Nonlinear Diff. Equ. Appl., 22 (2015), 651-672.
doi: 10.1007/s00030-014-0299-5. |
[19] |
J. Q. Liu, Y. Q. Wang and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Diff. Eq., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[20] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[21] |
X. Q. Liu, Y. S. Huang and J. Q. Liu,
Sign-changing solutions for an asymptotically linear Schrödinger equation with deepening potential well, Adv. Diff. Eq., 16 (2011), 1-30.
|
[22] |
M. Poppenberg, K. Schmitt and Z. Q. Wang,
On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var., 14 (2002), 329-344.
doi: 10.1007/s005260100105. |
[23] |
A. Selvitella,
Nondegeneracy of the ground state for quasilinear Schrödinger equations, Calc. Var., 53 (2015), 349-364.
doi: 10.1007/s00526-014-0751-8. |
[24] |
M. Struwe, Variational Methods, second ed., Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03212-1.![]() ![]() ![]() |
[25] |
C. A. Stuart, An introduction to elliptic equation on $\mathbb{R}^N$, in Nonlinear Functional Analysis and Applications to Differential Equations (A. Ambrosetti, K.-C. Chang and I. Ekeland eds.), World Scientific, Singapore, 1998. |
[26] |
C. A. Stuart and H. S. Zhou,
Applying the mountain pass theorem to an asymptotically linear elliptic equation on $\mathbb{R}^N$, Comm. Partial Diff. Eq., 24 (1999), 1731-1758.
doi: 10.1080/03605309908821481. |
[27] |
A. Szulkin and T. Weth,
Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.
doi: 10.1016/j.jfa.2009.09.013. |
[28] |
A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis
and Applications, Int. Press, (2010), 597-632. |
[29] |
M. Willem, Minimax Theorems, in Progress in Nonlinear Differential Equations and Their
Applications, 24 , Birkhäuser Boston, Inc., Boston, (1996), ⅹ-162.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
Y. J. Wang and W. M. Zou,
Bound states to critical quasilinear Schrödinger equations, Nonl. Diff. Eq. Appl., 19 (2012), 19-47.
doi: 10.1007/s00030-011-0116-3. |
[31] |
M. B. Yang and Y. H. Ding,
Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in $\mathbb{R}^N$, Ann. Mat. Pura Appl., 192 (2013), 783-804.
doi: 10.1007/s10231-011-0246-6. |
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