March  2019, 39(3): 1311-1343. doi: 10.3934/dcds.2019056

Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $

School of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author

Received  January 2018 Published  December 2018

Fund Project: Supported by NSFC (No.11371146), the Fundamental Research Funds for the Central Universities (No.2015zz133)

We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with subcritical or critical growth:
$ \begin{equation*}-Δ u+W(x)u^{2α-1}-ul'(u^2)Δ l(u^2) = f(u) \ \mbox{or}\ h(u)+u^{2^*-1},\ x∈\mathbb{R}^N,\end{equation*} $
where
$W(x):\mathbb{R}^N \to \mathbb{R} $
is a given potential and
$ l,h,f $
are real functions,
$ u>0,$
$ 2^* = 2N/(N-2), $
$ N≥3 $
. Our results cover physical models
$ l(s) = s^{\frac{α}{2}}, $
$ \frac{1}{2}<α<1. $
Citation: Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056
References:
[1]

S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324.   Google Scholar

[2]

S. Adachia and T. Watanable, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833.  doi: 10.1016/j.na.2011.09.015.  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. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[4]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[6]

J. M. Bezerra do ÓO. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Soblev exponent, Comm. Pure. Appl. Math, 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[8]

Y. K. Cheng and Y. T. Shen, Generalized quasilinear Schrödinger equations with critical growth, Appl. Math. Lett., 65 (2017), 106-112.  doi: 10.1016/j.aml.2016.10.011.  Google Scholar

[9]

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

[10]

D. G. Costa and C. A. Magalhães, Variational elliptic problems which are non-quadratic at infinity, Nonlinear. Anal. TMA., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

[11]

Y. B. DengS. J. Peng and S. S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[12]

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

[13]

W. Huang and J. Xiang, Soliton solutions for a quasilinear Schrödinger equation with critical exponent, Comm. Pure Appl. Anal., 15 (2016), 1309-1333.  doi: 10.3934/cpaa.2016.15.1309.  Google Scholar

[14]

L. Jeanjean and K. Tanaka, A remark on least energy solution in $ \mathbb{R}^{N} $, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[15]

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

[16]

G. Li, Positive solution for quasilinear Schrödinger Equations with a parameter, Comm. Pure. Appl. Anal,, 14 (2015), 1803-1816.  doi: 10.3934/cpaa.2015.14.1803.  Google Scholar

[17]

P. L. Lions, The concentration compactness principle in the calculus of variations, the locally compact case, parts Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non. Lineairé., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[18]

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

[19]

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

[20]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[21]

Y. T. Shen and X. K. Guo, Discussion of nontrivial critical points of the functional $\int_{\Omega }{F(x,u,Du)\text{d}x}$, Acta. Math. Sci., 10 (1990), 249-258(in Chinese).  Google Scholar

[22]

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

[23]

Y. T. Shen and Y. J. Wang, A class of generalized quasilinear Schrödinger equations, Comm. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.  Google Scholar

[24]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[25]

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

show all references

References:
[1]

S. Adachia and T. Watanable, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Diff. Eqns., 16 (2011), 289-324.   Google Scholar

[2]

S. Adachia and T. Watanable, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonl. Anal. TMA., 75 (2012), 819-833.  doi: 10.1016/j.na.2011.09.015.  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. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[4]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[6]

J. M. Bezerra do ÓO. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[7]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Soblev exponent, Comm. Pure. Appl. Math, 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[8]

Y. K. Cheng and Y. T. Shen, Generalized quasilinear Schrödinger equations with critical growth, Appl. Math. Lett., 65 (2017), 106-112.  doi: 10.1016/j.aml.2016.10.011.  Google Scholar

[9]

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

[10]

D. G. Costa and C. A. Magalhães, Variational elliptic problems which are non-quadratic at infinity, Nonlinear. Anal. TMA., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X.  Google Scholar

[11]

Y. B. DengS. J. Peng and S. S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[12]

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

[13]

W. Huang and J. Xiang, Soliton solutions for a quasilinear Schrödinger equation with critical exponent, Comm. Pure Appl. Anal., 15 (2016), 1309-1333.  doi: 10.3934/cpaa.2016.15.1309.  Google Scholar

[14]

L. Jeanjean and K. Tanaka, A remark on least energy solution in $ \mathbb{R}^{N} $, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[15]

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

[16]

G. Li, Positive solution for quasilinear Schrödinger Equations with a parameter, Comm. Pure. Appl. Anal,, 14 (2015), 1803-1816.  doi: 10.3934/cpaa.2015.14.1803.  Google Scholar

[17]

P. L. Lions, The concentration compactness principle in the calculus of variations, the locally compact case, parts Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non. Lineairé., 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[18]

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

[19]

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

[20]

X. Q. LiuJ. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[21]

Y. T. Shen and X. K. Guo, Discussion of nontrivial critical points of the functional $\int_{\Omega }{F(x,u,Du)\text{d}x}$, Acta. Math. Sci., 10 (1990), 249-258(in Chinese).  Google Scholar

[22]

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

[23]

Y. T. Shen and Y. J. Wang, A class of generalized quasilinear Schrödinger equations, Comm. Pure Appl. Anal., 15 (2016), 853-870.  doi: 10.3934/cpaa.2016.15.853.  Google Scholar

[24]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[25]

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

[1]

Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99

[2]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345

[3]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273

[4]

Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054

[5]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[6]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

[7]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[8]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345

[9]

Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025

[10]

Yi He, Gongbao Li. Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 731-762. doi: 10.3934/dcds.2016.36.731

[11]

Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237

[12]

Yaotian Shen, Youjun Wang. A class of generalized quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2016, 15 (3) : 853-870. doi: 10.3934/cpaa.2016.15.853

[13]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803

[14]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[15]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

[16]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

[17]

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

[18]

João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621

[19]

João Marcos do Ó, Abbas Moameni. Solutions for singular quasilinear Schrödinger equations with one parameter. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1011-1023. doi: 10.3934/cpaa.2010.9.1011

[20]

Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (85)
  • HTML views (63)
  • Cited by (0)

Other articles
by authors

[Back to Top]