Positive solutions for a critical quasilinear Schrödinger equation

Liang Xue; Jiafa Xu; Donal O'Regan; Liang Xue; Jiafa Xu; Donal O'Regan

doi:10.3934/math.2023998

AIMS Mathematics

2023, Volume 8, Issue 8: 19566-19581. doi: 10.3934/math.2023998

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Positive solutions for a critical quasilinear Schrödinger equation

1.
Department of Basic Courses, Anhui Industrial Polytechnic College, Tongling 244000, China
2.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
3.
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Received: 09 March 2023 Revised: 31 May 2023 Accepted: 06 June 2023 Published: 09 June 2023
MSC : 35J20, 35J70, 35P05, 35P30, 34B15, 58E05, 47H04

In our current work we investigate the following critical quasilinear Schrödinger equation

$ -\Delta \Theta+\mathcal V(x)\Theta-\Delta (\Theta^2)\Theta = |\Theta|^{22^*-2}\Theta+\lambda \mathcal K(x)g(\Theta), \ x \ \in \mathbb R^N, $

where $ N\geq 3 $, $ \lambda > 0 $, $ \mathcal V, \ \mathcal K\in C(\mathbb R^N, \mathbb R^+) $ and $ g\in C(\mathbb R, \mathbb R) $ has a quasicritical growth condition. We use the dual approach and the mountain pass theorem to show that the considered problem has a positive solution when $ \lambda $ is a large parameter.
- quasilinear Schrödinger equations,
- quasicritical growths,
- dual approaches
Citation: Liang Xue, Jiafa Xu, Donal O'Regan. Positive solutions for a critical quasilinear Schrödinger equation[J]. AIMS Mathematics, 2023, 8(8): 19566-19581. doi: 10.3934/math.2023998

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Abstract

In our current work we investigate the following critical quasilinear Schrödinger equation

$ -\Delta \Theta+\mathcal V(x)\Theta-\Delta (\Theta^2)\Theta = |\Theta|^{22^*-2}\Theta+\lambda \mathcal K(x)g(\Theta), \ x \ \in \mathbb R^N, $

where $ N\geq 3 $, $ \lambda > 0 $, $ \mathcal V, \ \mathcal K\in C(\mathbb R^N, \mathbb R^+) $ and $ g\in C(\mathbb R, \mathbb R) $ has a quasicritical growth condition. We use the dual approach and the mountain pass theorem to show that the considered problem has a positive solution when $ \lambda $ is a large parameter.

References

[1]	A. D. Bouard, N. Hayashi, J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73–105. https://doi.org/10.1007/s002200050191 doi: 10.1007/s002200050191
[2]	X. L. Chen, R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082–2085. https://doi.org/10.1103/PhysRevLett.70.2082 doi: 10.1103/PhysRevLett.70.2082
[3]	H. Lange, M. Poppenberg, H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Commun. Part. Diff. Eq., 24 (1999), 1399–1418. https://doi.org/10.1080/03605309908821469 doi: 10.1080/03605309908821469
[4]	M. Poppenberg, K. Schmitt, Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Dif., 14 (2002), 329–344. https://doi.org/10.1007/s005260100105 doi: 10.1007/s005260100105
[5]	J. Chen, X. Huang, B. Cheng, X. Tang, Existence and concentration behavior of ground state solutions for a class of generalized quasilinear Schrödinger equations in $\mathbb R^N$, Acta Math. Sci., 40B (2020), 1495–1524. https://doi.org/10.1007/s10473-020-0519-5 doi: 10.1007/s10473-020-0519-5
[6]	X. Zhang, L. Liu, Y. Wu, Y. Cui, Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach, Electron. J. Differ. Equ., 147 (2018), 1–15.
[7]	S. Chen, X. Wu, Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type, J. Math. Anal. Appl., 475 (2019), 1754–1777. https://doi.org/10.1016/j.jmaa.2019.03.051 doi: 10.1016/j.jmaa.2019.03.051
[8]	W. Zhu, C. Chen, Ground state sign-changing solutions for a class of quasilinear Schrödinger equations, Open Math., 19 (2021), 1746–1754. https://doi.org/10.1515/math-2021-0134 doi: 10.1515/math-2021-0134
[9]	K. Tu, Y. Cheng, On a class of quasilinear Schrödinger equations with the supercritical growth, J. Math. Phys., 62 (2021), 121508. https://doi.org/10.1063/5.0072312 doi: 10.1063/5.0072312
[10]	Y. Xue, L. Yu, J. Han, Existence of ground state solutions for generalized quasilinear Schrödinger equations with asymptotically periodic potential, Qual. Theor. Dyn. Syst., 21 (2022), 67. https://doi.org/10.1007/s12346-022-00590-1 doi: 10.1007/s12346-022-00590-1
[11]	Y. Wei, C. Chen, H. Yang, Z. Xiu, Existence and nonexistence of entire large solutions to a class of generalized quasilinear Schrödinger equations, Appl. Math. Lett., 133 (2002), 108296. https://doi.org/10.1016/j.aml.2022.108296 doi: 10.1016/j.aml.2022.108296
[12]	S. Zhang, Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations, AIMS Math., 7 (2021), 1015–1034. https://doi.org/10.3934/math.2022061 doi: 10.3934/math.2022061
[13]	Q. Jin, Standing wave solutions for a generalized quasilinear Schrödinger equation with indefinite potential, Appl. Anal., 2022. https://doi.org/10.1080/00036811.2022.2107907
[14]	W. Wang, Y. Zhang, Positive solutions for a relativistic nonlinear Schrödinger equation with critical exponent and Hardy potential, Complex Var. Elliptic, 67 (2022), 2924–2943. https://doi.org/10.1080/17476933.2021.1958798 doi: 10.1080/17476933.2021.1958798
[15]	J. Zhang, Multiple solutions for a quasilinear Schrödinger-Poisson system, Bound. Value Probl., 2021 (2021), 78. https://doi.org/10.1186/s13661-021-01553-2 doi: 10.1186/s13661-021-01553-2
[16]	A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117–144. https://doi.org/10.4171/JEMS/24 doi: 10.4171/JEMS/24
[17]	B. Opic, A. Kufner, Hardy-type inequalities, Pitman research notes in mathematics series, Longman Scientific & Technical, Harlow, 1990.
[18]	A. Ambrosetti, Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differ. Integral Equ., 18 (2005), 1321–1332.
[19]	D. Bonheure, J. Van Schaftingen, Ground states for nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pur. Appl., 189 (2010), 273–301. Available from: https://link.springer.com/article/10.1007/s10231-009-0109-6.
[20]	M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal., 56 (2004), 213–226. https://doi.org/10.1016/j.na.2003.09.008
[21]	J. Liu, Y. Wang, Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, II, J. Differ. Equ., 187 (2003), 473–493. https://doi.org/10.1016/S0022-0396(02)00064-5 doi: 10.1016/S0022-0396(02)00064-5
[22]	J. M. Bezerra do Ó, O. H. Miyagaki, S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 248 (2010), 722–744. https://doi.org/10.1016/j.jde.2009.11.030 doi: 10.1016/j.jde.2009.11.030
[23]	X. He, A. Qian, W. Zou, Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth, Nonlinearity, 26 (2013), 3137–3168. https://doi.org/10.1088/0951-7715/26/12/3137 doi: 10.1088/0951-7715/26/12/3137
[24]	E. Gloss, Existence and concentration of positive solutions for a quasilinear equation in $\mathbb{R}^N$, J. Math. Anal. Appl., 371 (2010), 465–484. https://doi.org/10.1016/j.jmaa.2010.05.033 doi: 10.1016/j.jmaa.2010.05.033
[25]	C. O. Alves, M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differ. Equ., 254 (2013), 1977–1991. https://doi.org/10.1016/j.jde.2012.11.013 doi: 10.1016/j.jde.2012.11.013
[26]	Q. Li, K. Teng, X. Wu, Existence of positive solutions for a class of critical fractional Schrödinger equations with potential vanishing at infinity, Mediterr. J. Math., 14 (2017), 80. https://doi.org/10.1007/s00009-017-0846-5 doi: 10.1007/s00009-017-0846-5
[27]	H. Berestycki, P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313–346. Available from: https://link.springer.com/article/10.1007/BF00250555.
[28]	E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935–2949. https://doi.org/10.1016/j.na.2009.11.037 doi: 10.1016/j.na.2009.11.037
[29]	E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Part. Dif., 39 (2010), 1–33. https://doi.org/10.1007/s00526-009-0299-1 doi: 10.1007/s00526-009-0299-1
[30]	X. Liu, J. Liu, Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Part. Dif., 46 (2013), 641–669. https://doi.org/10.1007/s00526-012-0497-0 doi: 10.1007/s00526-012-0497-0

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