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doi: 10.3934/dcdsb.2021214
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## Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

 1 College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, Yunnan, China 2 Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China

* Corresponding author: Guofa Li

Received  April 2021 Revised  June 2021 Early access August 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971339, 11901345), the Yunnan Local Colleges Applied Basic Research Projects (Grant No. 202001BA070001-032) and Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China (Grant No. 2020CXTD25)

In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations
 $\begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*}$
where
 $\kappa>0$
,
 $\lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R})$
is superlinear at infinity, the potentials
 $V(x)$
and
 $K(x)$
are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful
 $L^{\infty}$
-estimates. For the subcritical case (
 $\mu = 0$
) we can deal with large
 $\kappa>0$
. For the critical case we treat that
 $\kappa>0$
is small.
Citation: Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021214
##### References:
 [1] J. F. L. Aires and M. A. S. Souto, Equation with positive coefficient in the quasilinear term and vanishing potential, Topol. Methods Nonlinear Anal., 46 (2015), 813-833.   Google Scholar [2] C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar [3] C. O. Alves, Y. Wang and Y. 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] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations Ⅰ Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar [5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar [6] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.  doi: 10.1016/S0167-2789(01)00332-3.  Google Scholar [7] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.  doi: 10.1088/0951-7715/16/4/317.  Google Scholar [8] J. Chen, X. Huang and B. Cheng, Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition, Appl. Math. Lett., 87 (2019), 165-171.  doi: 10.1016/j.aml.2018.07.035.  Google Scholar [9] J. Chen, X. Huang, B. Cheng and C. Zhu, Some results on standing wave solutions for a class of quasilinear Schrödinger equations, J. Math. Phys., 60 (2019), 091506, 55 pp. doi: 10.1063/1.5093720.  Google Scholar [10] A. de Bouard, N. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar [11] Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 281-299.  doi: 10.1017/S0308210513001170.  Google Scholar [12] B. Hartmann and W. J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68 (2003), 184302. doi: 10.1103/PhysRevB.68.184302.  Google Scholar [13] C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Mathe. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.  Google Scholar [14] 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 [15] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.  Google Scholar [16] Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schödinger equations with critical growth, J. Math. Phys., 58 (2017), 041501. doi: 10.1063/1.4982035.  Google Scholar [17] Z. Liang, J. Gao and A. Li, Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities, J. Math. Anal. Appl., 484 (2020), 123732. doi: 10.1016/j.jmaa.2019.123732.  Google Scholar [18] P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar [19] U. B. Severo and G. M. de Carvalho, Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials, Appl. Anal., 100 (2021), 229-252.  doi: 10.1080/00036811.2019.1599106.  Google Scholar [20] U. B. Severo, E. Gloss and E. D. da Silva, On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms, J. Differential Equations, 263 (2017), 3550-3580.  doi: 10.1016/j.jde.2017.04.040.  Google Scholar [21] Y. Shen and Y. Wang, Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.  doi: 10.1080/17476933.2015.1119818.  Google Scholar [22] Y. Wang, A class of quasilinear Schrödinger equations with critical or supercritical exponents, Compu. Math. Appl., 70 (2015), 562-572.  doi: 10.1016/j.camwa.2015.05.016.  Google Scholar [23] Y. Wang and Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese Journal of Math., 22 (2018), 401-420.  doi: 10.11650/tjm/8150.  Google Scholar [24] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [25] M. Yang, C. A. Santos and J. Zhou, Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math., 21 (2019), 1850026, 23 pp. doi: 10.1142/S0219199718500268.  Google Scholar [26] W. Zou, Sign-Changing Critical Points Theory, Springer, New York, 2008.  Google Scholar

show all references

##### References:
 [1] J. F. L. Aires and M. A. S. Souto, Equation with positive coefficient in the quasilinear term and vanishing potential, Topol. Methods Nonlinear Anal., 46 (2015), 813-833.   Google Scholar [2] C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.  Google Scholar [3] C. O. Alves, Y. Wang and Y. 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] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations Ⅰ Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar [5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar [6] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.  doi: 10.1016/S0167-2789(01)00332-3.  Google Scholar [7] L. Brizhik, A. Eremko, B. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.  doi: 10.1088/0951-7715/16/4/317.  Google Scholar [8] J. Chen, X. Huang and B. Cheng, Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition, Appl. Math. Lett., 87 (2019), 165-171.  doi: 10.1016/j.aml.2018.07.035.  Google Scholar [9] J. Chen, X. Huang, B. Cheng and C. Zhu, Some results on standing wave solutions for a class of quasilinear Schrödinger equations, J. Math. Phys., 60 (2019), 091506, 55 pp. doi: 10.1063/1.5093720.  Google Scholar [10] A. de Bouard, N. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar [11] Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 281-299.  doi: 10.1017/S0308210513001170.  Google Scholar [12] B. Hartmann and W. J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68 (2003), 184302. doi: 10.1103/PhysRevB.68.184302.  Google Scholar [13] C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Mathe. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.  Google Scholar [14] 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 [15] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.  Google Scholar [16] Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schödinger equations with critical growth, J. Math. Phys., 58 (2017), 041501. doi: 10.1063/1.4982035.  Google Scholar [17] Z. Liang, J. Gao and A. Li, Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities, J. Math. Anal. Appl., 484 (2020), 123732. doi: 10.1016/j.jmaa.2019.123732.  Google Scholar [18] P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar [19] U. B. Severo and G. M. de Carvalho, Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials, Appl. Anal., 100 (2021), 229-252.  doi: 10.1080/00036811.2019.1599106.  Google Scholar [20] U. B. Severo, E. Gloss and E. D. da Silva, On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms, J. Differential Equations, 263 (2017), 3550-3580.  doi: 10.1016/j.jde.2017.04.040.  Google Scholar [21] Y. Shen and Y. Wang, Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.  doi: 10.1080/17476933.2015.1119818.  Google Scholar [22] Y. Wang, A class of quasilinear Schrödinger equations with critical or supercritical exponents, Compu. Math. Appl., 70 (2015), 562-572.  doi: 10.1016/j.camwa.2015.05.016.  Google Scholar [23] Y. Wang and Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese Journal of Math., 22 (2018), 401-420.  doi: 10.11650/tjm/8150.  Google Scholar [24] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [25] M. Yang, C. A. Santos and J. Zhou, Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math., 21 (2019), 1850026, 23 pp. doi: 10.1142/S0219199718500268.  Google Scholar [26] W. Zou, Sign-Changing Critical Points Theory, Springer, New York, 2008.  Google Scholar
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