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May  2021, 14(5): 1779-1799. 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  May 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

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

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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