June  2020, 19(6): 3429-3444. doi: 10.3934/cpaa.2020059

Existence results for quasilinear Schrödinger equations with a general nonlinearity

1. 

College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China

2. 

Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China

3. 

School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China

*Corresponding author

Received  January 2019 Revised  July 2019 Published  March 2020

Fund Project: H. Liu is supported by National Natural Science Foundation of China (No.11701220, No. 11926334, No.11926335). L. Zhao is supported by National Natural Science Foundation of China (No.11671026, No.11771385) and Beijing Municipal Commission of Education KZ202010028048

Consider the quasilinear Schrödinger equation
$ \begin{equation*} \label{eq0.1}-\Delta u+V(x)u- \Delta(u^2)u = h(u)\ \ \mbox{in}\ {\mathbb{R}}^N,\tag{A} \end{equation*} $
where
$ N\geq 3 $
,
$ V: {\mathbb{R}}^N\to{\mathbb{R}} $
and
$ h: {\mathbb{R}}\to{\mathbb{R}} $
are functions. Under some general assumptions on
$ V $
and
$ h $
, we establish two existence results for problem (A) by using variational methods. The main novelty is that, unlike most other papers on this problem, we do not assume the nonlinear term to be 4-superlinear at infinity.
Citation: Haidong Liu, Leiga Zhao. Existence results for quasilinear Schrödinger equations with a general nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3429-3444. doi: 10.3934/cpaa.2020059
References:
[1]

S. AdachiM. Shibata and T. Watanabe, Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.  doi: 10.1016/j.jde.2013.11.004.  Google Scholar

[2]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324.   Google Scholar

[3]

C. O. AlvesY. J. Wang and Y. T. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[4]

D. ArcoyaL. Boccardo and L. Orsina, Critical points for functionals with quasilinear singular Euler-Lagrange equations, Calc. Var. Partial Differ. Equ., 47 (2013), 159-180.  doi: 10.1007/s00526-012-0514-3.  Google Scholar

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A. Azzollini and A. Pomponio, On the Schrödinger equation in ${\mathbb{R}}^N$ under the effect of a general nonlinear term, Indiana Univ. Math. J., 58 (2009), 1361-1378.  doi: 10.1512/iumj.2009.58.3576.  Google Scholar

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M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. Theory Methods Appl., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

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Y. B. DengS. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

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J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differ. Equ., 38 (2010), 275-315.  doi: 10.1007/s00526-009-0286-6.  Google Scholar

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X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254 (2013), 2015-2032.  doi: 10.1016/j.jde.2012.11.017.  Google Scholar

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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

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Y. X. Guo and Z. W. Tang, Multi-bump bound state solutions for the quasilinear Schrödinger equation with critical frequency, Pac. J. Math., 270 (2014), 49-77.  doi: 10.2140/pjm.2014.270.49.  Google Scholar

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X. M. HeA. X. Qian and W. M. Zou, Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

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L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbb{R}}^N$, Proc. R. Soc. Edinb. Sect. A Math., 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

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L. Jeanjean, Local conditions insuring bifurcation from the continuous spectrum, Math. Z., 232 (1999), 651-664.  doi: 10.1007/PL00004774.  Google Scholar

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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.  Google Scholar

[18]

Y. T. Jing, Z. L. Liu and Z. Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differ. Equ., 55 (2016), 150. doi: 10.1007/s00526-016-1067-7.  Google Scholar

[19]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[20]

E. W. LaedkeK. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.  doi: 10.1063/1.525675.  Google Scholar

[21]

J. Q. LiuX. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differ. Equ., 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[22]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[23]

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

[24]

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

[25]

J. Q. Liu and Z. Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differ. Equ., 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar

[26]

J. Q. LiuZ. Q. Wang and Y. X. Guo, Multibump solutions for quasilinear elliptic equations, J. Funct. Anal., 262 (2012), 4040-4102.  doi: 10.1016/j.jfa.2012.02.009.  Google Scholar

[27]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[28]

V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. Rev. Sec. Phys. Lett., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[29]

M. PoppenbergK. Schmitt and Z. Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[30]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.  doi: 10.1088/0951-7715/23/5/011.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

show all references

References:
[1]

S. AdachiM. Shibata and T. Watanabe, Blow-up phenomena and asymptotic profiles of ground states of quasilinear elliptic equations with $H^1$-supercritical nonlinearities, J. Differ. Equ., 256 (2014), 1492-1514.  doi: 10.1016/j.jde.2013.11.004.  Google Scholar

[2]

S. Adachi and T. Watanabe, $G$-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equ., 16 (2011), 289-324.   Google Scholar

[3]

C. O. AlvesY. J. Wang and Y. T. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[4]

D. ArcoyaL. Boccardo and L. Orsina, Critical points for functionals with quasilinear singular Euler-Lagrange equations, Calc. Var. Partial Differ. Equ., 47 (2013), 159-180.  doi: 10.1007/s00526-012-0514-3.  Google Scholar

[5]

A. Azzollini and A. Pomponio, On the Schrödinger equation in ${\mathbb{R}}^N$ under the effect of a general nonlinear term, Indiana Univ. Math. J., 58 (2009), 1361-1378.  doi: 10.1512/iumj.2009.58.3576.  Google Scholar

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅰ, Existence of a ground state, Arch. Ration. Meth. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[8]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. Theory Methods Appl., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[9]

Y. B. DengS. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.  doi: 10.1016/j.jde.2015.09.021.  Google Scholar

[10]

J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differ. Equ., 38 (2010), 275-315.  doi: 10.1007/s00526-009-0286-6.  Google Scholar

[11]

X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254 (2013), 2015-2032.  doi: 10.1016/j.jde.2012.11.017.  Google Scholar

[12]

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

[13]

Y. X. Guo and Z. W. Tang, Multi-bump bound state solutions for the quasilinear Schrödinger equation with critical frequency, Pac. J. Math., 270 (2014), 49-77.  doi: 10.2140/pjm.2014.270.49.  Google Scholar

[14]

X. M. HeA. X. Qian and W. M. Zou, Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

[15]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbb{R}}^N$, Proc. R. Soc. Edinb. Sect. A Math., 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar

[16]

L. Jeanjean, Local conditions insuring bifurcation from the continuous spectrum, Math. Z., 232 (1999), 651-664.  doi: 10.1007/PL00004774.  Google Scholar

[17]

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.  Google Scholar

[18]

Y. T. Jing, Z. L. Liu and Z. Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differ. Equ., 55 (2016), 150. doi: 10.1007/s00526-016-1067-7.  Google Scholar

[19]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn., 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[20]

E. W. LaedkeK. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.  doi: 10.1063/1.525675.  Google Scholar

[21]

J. Q. LiuX. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differ. Equ., 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[22]

J. Q. LiuY. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[23]

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

[24]

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

[25]

J. Q. Liu and Z. Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differ. Equ., 257 (2014), 2874-2899.  doi: 10.1016/j.jde.2014.06.002.  Google Scholar

[26]

J. Q. LiuZ. Q. Wang and Y. X. Guo, Multibump solutions for quasilinear elliptic equations, J. Funct. Anal., 262 (2012), 4040-4102.  doi: 10.1016/j.jfa.2012.02.009.  Google Scholar

[27]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[28]

V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. Rev. Sec. Phys. Lett., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[29]

M. PoppenbergK. Schmitt and Z. Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[30]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.  doi: 10.1088/0951-7715/23/5/011.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

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