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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[25]

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

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. 

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[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.

[25]

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

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