March  2022, 21(3): 1071-1091. doi: 10.3934/cpaa.2022010

Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, China

* Corresponding author

Received  March 2021 Revised  October 2021 Published  March 2022 Early access  December 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (No:11971485) and the NSFC (11871475)

In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:
$\left (1\!+\!b\int_{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2} dx \right) \left[-\mathrm{div} \left(g^{2}(u)\nabla u\right)\!+\!g(u)g'(u)|\nabla u|^{2}\right] \!+\!V(x)u\! = \!f( u),(\rm P)$
where
$ b>0 $
is a parameter,
$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $
,
$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $
and
$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $
. Under some "Berestycki-Lions type assumptions" on the nonlinearity
$ f $
which are almost necessary, we prove that problem
$ (\rm P) $
has a nontrivial solution
$ \bar{u}\in H^{1}(\mathbb{R}^{3}) $
such that
$ \bar{v} = G(\bar{u}) $
is a ground state solution of the following problem
$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} dx \right) \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$
where
$ G(t): = \int_{0}^{t} g(s) ds $
. We also give a minimax characterization for the ground state solution
$ \bar{v} $
.
Citation: Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1071-1091. doi: 10.3934/cpaa.2022010
References:
[1]

H. Berestycki and P. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[2]

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.

[3]

S. Cuccagna, On instability of excited states of the nonlinear quasilinear Schrödinger equation, Phys. D., 238 (2009), 38-54.  doi: 10.1016/j.physd.2008.08.010.

[4]

S. Chen and X. Tang, Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496-515.  doi: 10.1515/anona-2020-0011.

[5]

J. ChenX. TangZ. Gao and B. Cheng, Ground state sign-changing solutions for a class of generelized quasilinear Schrödinger equations with Kirchhoff-type perturbation, J. Fixed Point Theory Appl., 19 (2017), 3127-3149.  doi: 10.1007/s11784-017-0475-4.

[6]

M. Colin and L. Jeanjean, Louis solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[7]

J. ChenX. Tang and B. Cheng, Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving a Kirchhoff-type perturbation with critical Sobolev exponent, J. Math. Phys., 59 (2018), 021505.  doi: 10.1063/1.5024898.

[8]

Y. DengS. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 260 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.

[9]

Y. DengW. Huang and S. Zhang, Ground state solutions for generalized quasilinear Schrödinger equations with critical growth and lower power subcritical perturbation, Adv. Nonlinear Stud., 19 (2019), 219-237.  doi: 10.1515/ans-2018-2029.

[10]

Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259 (2015), 2884-2902.  doi: 10.1016/j.jde.2015.04.005.

[11]

D. Hu, X. Tang and Q. Zhang, Existence of ground state solutions for Kirchhoff-type problem with variable potential, Appl. Anal., (2021), 1–14. doi: 10.1080/00036811.2021.1947499.

[12]

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

[13]

L. Jeanjean and J. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 23-28.  doi: 10.1016/S0764-4442(98)80097-9.

[14]

G. Kirchhoff, Mechanik, Teubner, Leipzig., 1883.

[15]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3}$, J. Differ. Equ., 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.

[16]

F. LiX. Zhu and Z. Liang, Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl., 443 (2016), 11-38.  doi: 10.1016/j.jmaa.2016.05.005.

[17]

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

[18]

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

[19]

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

[20]

P. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part I, Ann Inst H Poincaré Anal Non Linéaire, 1 (1984), 109-145. 

[21]

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

[22]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.

[23]

X. Tang and S. Chen, Ground stste solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differ. Equ., 56 (2017), 110-134.  doi: 10.1007/s00526-017-1214-9.

[24]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

J. Zhao and X. Liu, Ground state solutions for quasilinear equations of Kirchhoff type, J. Differ. Equ., 2020 (2020), 1-14. 

[26]

Q. Zhang and D. Hu, Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type, Complex Var. Elliptic Equ., (2021), 1–15. doi: 10.1080/17476933.2021.1916918.

[27]

J. Zhang, X. Tang and D. Qin, Infinitely many solutions for Kirchhoff problems with lack of compactness, Nonlinear Anal., 197 (2020), 111856, 31 pp. doi: 10.1016/j.na.2020.111856.

show all references

References:
[1]

H. Berestycki and P. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.

[2]

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.

[3]

S. Cuccagna, On instability of excited states of the nonlinear quasilinear Schrödinger equation, Phys. D., 238 (2009), 38-54.  doi: 10.1016/j.physd.2008.08.010.

[4]

S. Chen and X. Tang, Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496-515.  doi: 10.1515/anona-2020-0011.

[5]

J. ChenX. TangZ. Gao and B. Cheng, Ground state sign-changing solutions for a class of generelized quasilinear Schrödinger equations with Kirchhoff-type perturbation, J. Fixed Point Theory Appl., 19 (2017), 3127-3149.  doi: 10.1007/s11784-017-0475-4.

[6]

M. Colin and L. Jeanjean, Louis solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.

[7]

J. ChenX. Tang and B. Cheng, Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving a Kirchhoff-type perturbation with critical Sobolev exponent, J. Math. Phys., 59 (2018), 021505.  doi: 10.1063/1.5024898.

[8]

Y. DengS. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 260 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.

[9]

Y. DengW. Huang and S. Zhang, Ground state solutions for generalized quasilinear Schrödinger equations with critical growth and lower power subcritical perturbation, Adv. Nonlinear Stud., 19 (2019), 219-237.  doi: 10.1515/ans-2018-2029.

[10]

Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differ. Equ., 259 (2015), 2884-2902.  doi: 10.1016/j.jde.2015.04.005.

[11]

D. Hu, X. Tang and Q. Zhang, Existence of ground state solutions for Kirchhoff-type problem with variable potential, Appl. Anal., (2021), 1–14. doi: 10.1080/00036811.2021.1947499.

[12]

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

[13]

L. Jeanjean and J. Toland, Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 23-28.  doi: 10.1016/S0764-4442(98)80097-9.

[14]

G. Kirchhoff, Mechanik, Teubner, Leipzig., 1883.

[15]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3}$, J. Differ. Equ., 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.

[16]

F. LiX. Zhu and Z. Liang, Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl., 443 (2016), 11-38.  doi: 10.1016/j.jmaa.2016.05.005.

[17]

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

[18]

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

[19]

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

[20]

P. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case. Part I, Ann Inst H Poincaré Anal Non Linéaire, 1 (1984), 109-145. 

[21]

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

[22]

Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.  doi: 10.1016/j.na.2012.10.005.

[23]

X. Tang and S. Chen, Ground stste solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differ. Equ., 56 (2017), 110-134.  doi: 10.1007/s00526-017-1214-9.

[24]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

J. Zhao and X. Liu, Ground state solutions for quasilinear equations of Kirchhoff type, J. Differ. Equ., 2020 (2020), 1-14. 

[26]

Q. Zhang and D. Hu, Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type, Complex Var. Elliptic Equ., (2021), 1–15. doi: 10.1080/17476933.2021.1916918.

[27]

J. Zhang, X. Tang and D. Qin, Infinitely many solutions for Kirchhoff problems with lack of compactness, Nonlinear Anal., 197 (2020), 111856, 31 pp. doi: 10.1016/j.na.2020.111856.

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