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January  2019, 18(1): 51-64. doi: 10.3934/cpaa.2019004

A positive solution for an asymptotically cubic quasilinear Schrödinger equation

Xiang-Dong Fang

School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China

Received  August 2017 Revised  April 2018 Published  August 2018

Fund Project: This project is supported by National Natural Science Foundation of China (Grant No. 11601057) and the Fundamental Research Funds for the Central Universities (Grant. DUT18LK05).

We consider the following quasilinear Schrödinger equation
$ - \Delta u + V(x)u - \Delta ({u^2})u = q(x)g(u),\;\;\;\;x \in {\mathbb{R}^N}, $
where
$N≥ 1$
,
$0 < q(x)≤ \lim_{|x|\to∞}q(x)$
,
$g∈ C(\mathbb{R}^+, \mathbb{R})$
 and
$g(u)/u^3 \to 1$
, as
$u \to ∞.$
 We establish the existence of a positive solution to this problem by using the method developed in Szulkin and Weth [27,28].
Keywords: Quasilinear Schrödinger equation, positive solution, asymptotically cubic, Nehari manifold.
Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 49J35.
Citation: Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var., 17 (2003), 257-281.   Google Scholar

[7] K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] M. Struwe, Variational Methods, second ed., Springer-Verlag, Berlin, 1996.  doi: 10.1007/978-3-662-03212-1.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, (2010), 597-632.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var., 17 (2003), 257-281.   Google Scholar

[7] K. C. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] M. Struwe, Variational Methods, second ed., Springer-Verlag, Berlin, 1996.  doi: 10.1007/978-3-662-03212-1.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, Int. Press, (2010), 597-632.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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