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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  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, 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.  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

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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

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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

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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

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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

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