# American Institute of Mathematical Sciences

May  2021, 14(5): 1779-1799. doi: 10.3934/dcdss.2020454

## Sign-changing solutions for a parameter-dependent quasilinear equation

 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.
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.

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##### 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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