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September  2017, 16(5): 1603-1615. doi: 10.3934/cpaa.2017077

Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$

School of Mathematical Sciences, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024 Dalian, PR China

Received  August 2016 Revised  March 2017 Published  May 2017

Fund Project: The author is supported by Fundamental Research Funds for the Central Universities grant DUT16RC(4)54 and DUT15QY20, China Postdoctoral Science Foundation grant 2015M571293 and National Natural Science Foundation of China grant 11601057.

In this article we study the following quasilinear Schrödinger equation
$-Δ u+V(x)u-Δ(u^{2})u=g(u), x∈ \mathbb{R}^{N},$
where
$ V(x)$
tends to some limit
$V_{∞}>0 $
as
$|x|\to∞ $
and
$g∈ C(\mathbb{R},\mathbb{R}) $
. We prove the existence of positive solutions by using the Nehari manifold.
Citation: Xiang-Dong Fang. Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1603-1615. doi: 10.3934/cpaa.2017077
References:
[1]

S. Adachi and T. Watanabe, Asymptotic properties of ground states of quasilinear Schrödinger equations with H1-subcritical exponent, Adv. Nonlinear Stud., 12 (2012), 255-279.  doi: 10.1515/ans-2012-0205.  Google Scholar

[2]

S. Adachi and T. Watanabe, Asymptotic uniqueness of ground states for a class of quasilinear Schrödinger equations with H1-supercritical exponent, J. Diff. Eq., 260 (2016), 3086-3118.  doi: 10.1016/j.jde.2015.10.029.  Google Scholar

[3]

A. AmbrosettiG. 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]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.  Google Scholar

[5]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+1)/(N-2)} $ in $\mathbb{R}^N $, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

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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

[7]

P. C. CarriãoR. 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

[8]

G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74 (2006), 47-77.  doi: 10.1007/s00032-006-0059-z.  Google Scholar

[9]

G. Cerami and C. Maniscalco, Multiple positive solutions for a singularly perturbed Dirichlet problem in "geometrically trivial" domains, Topol. Methods Nonlinear Anal., 19 (2002), 63-76.  doi: 10.12775/TMNA.2002.004.  Google Scholar

[10]

G. Cerami and D. Passaseo, High energy positive solutions for mixed and Neumann elliptic problems with critical nonlinearity, J. Anal. Math., 71 (1997), 1-39.  doi: 10.1007/BF02788020.  Google Scholar

[11]

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

[12]

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

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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

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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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, in Cambridge University Press, (1993), xviii-258.  doi: 10.1017/CBO9780511551703.  Google Scholar

[20]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N $, in Mathematical Analysis and Applications (Part A), Academic Press, (1981), 369-402.   Google Scholar

[21]

E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbb{R}^N $, J. Math. Anal. Appl., 371 (2010), 465-484.  doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar

[22]

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

[23]

R. LehrerL. 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

[24]

J. Q. LiuY. 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

[25]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[26]

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

[27]

L. A. Maia and R. Ruviaro, Sign-changing solutions for a Schrödinger equation with saturable nonlinearity, Milan J. Math., 79 (2011), 259-271.  doi: 10.1007/s00032-011-0145-8.  Google Scholar

[28]

M. PoppenbergK. 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

[29]

E. A. Silva and G. G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonl. Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[30]

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

[31]

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

[32]

M. Willem, Minimax Theorems, in Progress in Nonlinear Differential Equations and their Applications, 24, Birkhࢴuser Boston, Inc., Boston, (1996), ⅹ-162.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[33]

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

show all references

References:
[1]

S. Adachi and T. Watanabe, Asymptotic properties of ground states of quasilinear Schrödinger equations with H1-subcritical exponent, Adv. Nonlinear Stud., 12 (2012), 255-279.  doi: 10.1515/ans-2012-0205.  Google Scholar

[2]

S. Adachi and T. Watanabe, Asymptotic uniqueness of ground states for a class of quasilinear Schrödinger equations with H1-supercritical exponent, J. Diff. Eq., 260 (2016), 3086-3118.  doi: 10.1016/j.jde.2015.10.029.  Google Scholar

[3]

A. AmbrosettiG. 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]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.  Google Scholar

[5]

V. Benci and G. Cerami, Existence of positive solutions of the equation $-Δ u+a(x)u=u^{(N+1)/(N-2)} $ in $\mathbb{R}^N $, J. Funct. Anal., 88 (1990), 90-117.  doi: 10.1016/0022-1236(90)90120-A.  Google Scholar

[6]

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

[7]

P. C. CarriãoR. 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

[8]

G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math., 74 (2006), 47-77.  doi: 10.1007/s00032-006-0059-z.  Google Scholar

[9]

G. Cerami and C. Maniscalco, Multiple positive solutions for a singularly perturbed Dirichlet problem in "geometrically trivial" domains, Topol. Methods Nonlinear Anal., 19 (2002), 63-76.  doi: 10.12775/TMNA.2002.004.  Google Scholar

[10]

G. Cerami and D. Passaseo, High energy positive solutions for mixed and Neumann elliptic problems with critical nonlinearity, J. Anal. Math., 71 (1997), 1-39.  doi: 10.1007/BF02788020.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, in Cambridge University Press, (1993), xviii-258.  doi: 10.1017/CBO9780511551703.  Google Scholar

[20]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N $, in Mathematical Analysis and Applications (Part A), Academic Press, (1981), 369-402.   Google Scholar

[21]

E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbb{R}^N $, J. Math. Anal. Appl., 371 (2010), 465-484.  doi: 10.1016/j.jmaa.2010.05.033.  Google Scholar

[22]

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

[23]

R. LehrerL. 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

[24]

J. Q. LiuY. 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

[25]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[26]

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

[27]

L. A. Maia and R. Ruviaro, Sign-changing solutions for a Schrödinger equation with saturable nonlinearity, Milan J. Math., 79 (2011), 259-271.  doi: 10.1007/s00032-011-0145-8.  Google Scholar

[28]

M. PoppenbergK. 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

[29]

E. A. Silva and G. G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonl. Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.  Google Scholar

[30]

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

[31]

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

[32]

M. Willem, Minimax Theorems, in Progress in Nonlinear Differential Equations and their Applications, 24, Birkhࢴuser Boston, Inc., Boston, (1996), ⅹ-162.  doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[33]

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

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