American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions
  • DCDS-S Home
  • This Issue
  • Next Article
    Solving a class of biological HIV infection model of latently infected cells using heuristic approach
doi: 10.3934/dcdss.2020454

Sign-changing solutions for a parameter-dependent quasilinear equation

Jiaquan Liu 1, , Xiangqing Liu 2,, and  Zhi-Qiang Wang 3,,

1. 

LMAM, School of Mathematical Science, Peking University, Beijing 100871, China
2. 

Department of Mathematics, Yunnan Normal University, Kunming 650500, China
3. 

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

* Corresponding author: Xiangqing Liu, Zhi-Qiang Wang

Received  February 2020 Revised  July 2020 Published  November 2020

We consider quasilinear elliptic equations, including the following Modified Nonlinear Schrödinger Equation as a special example:
$ \begin{equation*} \left\{ \begin{aligned} &\Delta u+\frac{1}{2}u\Delta u^2+\lambda |u|^{r-2}u = 0, \ \ \ \text{in}\,\,\Omega,\\ &u = 0\quad\text{on}\,\,\partial\Omega, \end{aligned} \right. \end{equation*} $
where
$ \Omega\subset\mathbb{R}^N(N\geq3) $
 is a bounded domain with smooth boundary,
$ \lambda>0,\, r\in(2,4) $
. We prove as
$ \lambda $
 becomes large the existence of more and more sign-changing solutions of both positive and negative energies.
Keywords: Quasilinear elliptic equation, sign-changing solutions, truncation techniques.
Mathematics Subject Classification: 35B20, 35H30.
Citation: Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020454
References:
[1]

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

[2]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.  doi: 10.1007/s002090050492.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.  doi: 10.1081/PDE-120028842.  Google Scholar

[4]

T. Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131.  doi: 10.12775/TMNA.1996.005.  Google Scholar

[5]

F. G. Bass and N. N. Nasonov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[7]

D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972/1973), 65-74.  doi: 10.1512/iumj.1973.22.22008.  Google Scholar

[8]

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

[9]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[10]

Y. Jing, Z. Liu and Z.-Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differential Equations, 55 (2016), 150, 26 pp. doi: 10.1007/s00526-016-1067-7.  Google Scholar

[11]

Y. Jing, Z. Liu and Z.-Q. Wang, Existence results for a singular quasilinear elliptic equation, J. Fixed Point Theory Appl., 19 (2017), 67-84.  doi: 10.1007/s11784-016-0341-9.  Google Scholar

[12]

Y. Jing, Z. Liu and Z.-Q. Wang, Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations, preprint. Google Scholar

[13]

M. Kosevich, A. Ivanov and S. Kovalev, Magnetic solutions, Phys. Rep., 194 (1990), 117-238.   Google Scholar

[14]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jap., 50 (1981), 3801-3805.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[15] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.   Google Scholar
[16]

S. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.  doi: 10.1090/S0002-9947-02-03031-3.  Google Scholar

[17]

G. M. Lieberman, The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.  Google Scholar

[18]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JEPT Letters, 27 (1978), 517-520.   Google Scholar

[19]

J.-Q. Liu, X.-Q. Liu and Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differential Equations, 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[20]

J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.  Google Scholar

[21]

J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar

[22]

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

[23]

X.-Q. Liu, J.-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

[24]

X. Liu, J. Liu and Z.-Q. Wang, Localized nodal solutions for quasilinear Schrödinger equations, J. Differential Equations, 267 (2019), 7411-7461.  doi: 10.1016/j.jde.2019.08.003.  Google Scholar

[25]

Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.  doi: 10.1006/jdeq.2000.3867.  Google Scholar

[26]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[27]

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

[28]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[29]

X. Liu and J. Zhao, $p$-Laplacian equation in $\mathbb{R}^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.  doi: 10.1515/ans-2015-5059.  Google Scholar

[30]

V. G. Makhan'kov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[31]

M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.  doi: 10.1063/1.861553.  Google Scholar

[32]

G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.  doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[33]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference, Series in Mathematics, American Mathematical Society, Vol. 65, 1986. doi: 10.1090/cbms/065.  Google Scholar

[34]

J. Zhao, X. Liu and J. Liu, $p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.  doi: 10.1016/j.jmaa.2017.03.085.  Google Scholar

show all references

References:
[1]

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

[2]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677.  doi: 10.1007/s002090050492.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Commun. Partial Differential Equations, 29 (2004), 25-42.  doi: 10.1081/PDE-120028842.  Google Scholar

[4]

T. Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal., 7 (1996), 115-131.  doi: 10.12775/TMNA.1996.005.  Google Scholar

[5]

F. G. Bass and N. N. Nasonov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.  doi: 10.1016/0370-1573(90)90093-H.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[7]

D. C. Clark, A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J., 22 (1972/1973), 65-74.  doi: 10.1512/iumj.1973.22.22008.  Google Scholar

[8]

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

[9]

R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.  doi: 10.1007/BF01325508.  Google Scholar

[10]

Y. Jing, Z. Liu and Z.-Q. Wang, Multiple solutions of a parameter-dependent quasilinear elliptic equation, Calc. Var. Partial Differential Equations, 55 (2016), 150, 26 pp. doi: 10.1007/s00526-016-1067-7.  Google Scholar

[11]

Y. Jing, Z. Liu and Z.-Q. Wang, Existence results for a singular quasilinear elliptic equation, J. Fixed Point Theory Appl., 19 (2017), 67-84.  doi: 10.1007/s11784-016-0341-9.  Google Scholar

[12]

Y. Jing, Z. Liu and Z.-Q. Wang, Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations, preprint. Google Scholar

[13]

M. Kosevich, A. Ivanov and S. Kovalev, Magnetic solutions, Phys. Rep., 194 (1990), 117-238.   Google Scholar

[14]

S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jap., 50 (1981), 3801-3805.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[15] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.   Google Scholar
[16]

S. Li and Z.-Q. Wang, Ljusternik-Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 3207-3227.  doi: 10.1090/S0002-9947-02-03031-3.  Google Scholar

[17]

G. M. Lieberman, The natural generalizationj of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differential Equations, 16 (1991), 311-361.  doi: 10.1080/03605309108820761.  Google Scholar

[18]

A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JEPT Letters, 27 (1978), 517-520.   Google Scholar

[19]

J.-Q. Liu, X.-Q. Liu and Z.-Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Commun. Partial Differential Equations, 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[20]

J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.  doi: 10.1016/j.jde.2016.09.018.  Google Scholar

[21]

J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.  doi: 10.1007/s00526-014-0724-y.  Google Scholar

[22]

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

[23]

X.-Q. Liu, J.-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

[24]

X. Liu, J. Liu and Z.-Q. Wang, Localized nodal solutions for quasilinear Schrödinger equations, J. Differential Equations, 267 (2019), 7411-7461.  doi: 10.1016/j.jde.2019.08.003.  Google Scholar

[25]

Z. Liu and J. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations, 172 (2001), 257-299.  doi: 10.1006/jdeq.2000.3867.  Google Scholar

[26]

J. Liu and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc., 131 (2003), 441-448.  doi: 10.1090/S0002-9939-02-06783-7.  Google Scholar

[27]

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

[28]

J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[29]

X. Liu and J. Zhao, $p$-Laplacian equation in $\mathbb{R}^N$ with finite potential via the truncation method, Adv. Nonlinear Stud., 17 (2017), 595-610.  doi: 10.1515/ans-2015-5059.  Google Scholar

[30]

V. G. Makhan'kov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1-86.  doi: 10.1016/0370-1573(84)90106-6.  Google Scholar

[31]

M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.  doi: 10.1063/1.861553.  Google Scholar

[32]

G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A, 110 (1982), 41-80.  doi: 10.1016/0378-4371(82)90104-2.  Google Scholar

[33]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference, Series in Mathematics, American Mathematical Society, Vol. 65, 1986. doi: 10.1090/cbms/065.  Google Scholar

[34]

J. Zhao, X. Liu and J. Liu, $p$-Laplacian equations in $\mathbb{R}^N$ with finite potential via truncation method, the critical case, J. Math. Anal. Appl., 455 (2017), 58-88.  doi: 10.1016/j.jmaa.2017.03.085.  Google Scholar

[1]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[2]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[3]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[4]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354
[5]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[6]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045
[7]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[8]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[9]

Andrea Malchiodi. Perturbative techniques for the construction of spike-layers. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3767-3787. doi: 10.3934/dcds.2020055
[10]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[11]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[12]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[13]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318
[14]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[15]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[16]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[17]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[18]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[19]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[20]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (25)
  • HTML views (95)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences