American Institute of Mathematical Sciences

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

Existence results for quasilinear Schrödinger equations with a general nonlinearity

Haidong Liu 1, and  Leiga Zhao 2,3,,

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.
Keywords: Quasilinear Schrödinger equation, well potential, radial potential, general nonlinearity.
Mathematics Subject Classification: Primary: 35J10; Secondary: 35J20, 35J60.
Citation: Haidong Liu, Leiga Zhao. Existence results for quasilinear Schrödinger equations with a general nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3429-3444. doi: 10.3934/cpaa.2020059
References:
[1]

S. Adachi, M. 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.

[2]

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

[3]

C. O. Alves, Y. 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.

[4]

D. Arcoya, L. 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.

[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.

[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.

[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.

[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.

[9]

Y. B. Deng, S. 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.

[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.

[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.

[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.

[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.

[14]

X. M. He, A. 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.

[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.

[16]

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

[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.

[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.

[19]

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

[20]

E. W. Laedke, K. 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.

[21]

J. Q. Liu, X. 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.

[22]

J. Q. Liu, Y. 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.

[23]

J. Q. Liu, Y. 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.

[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.

[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.

[26]

J. Q. Liu, Z. 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.

[27]

X. Q. Liu, J. 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.

[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.

[29]

M. Poppenberg, K. 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.

[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.

[31]

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

show all references

References:
[1]

S. Adachi, M. 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.

[2]

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

[3]

C. O. Alves, Y. 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.

[4]

D. Arcoya, L. 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.

[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.

[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.

[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.

[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.

[9]

Y. B. Deng, S. 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.

[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.

[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.

[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.

[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.

[14]

X. M. He, A. 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.

[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.

[16]

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

[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.

[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.

[19]

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

[20]

E. W. Laedke, K. 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.

[21]

J. Q. Liu, X. 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.

[22]

J. Q. Liu, Y. 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.

[23]

J. Q. Liu, Y. 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.

[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.

[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.

[26]

J. Q. Liu, Z. 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.

[27]

X. Q. Liu, J. 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.

[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.

[29]

M. Poppenberg, K. 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.

[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.

[31]

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

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