In this paper, critical point theory is used to show the existence of nontrivial solutions for a class of generalized quasilinear Schrödinger equations
$ \begin{equation*} -\Delta_pu-{|u|}^{\sigma-2}uh'({|u|}^\sigma)\Delta_ph({|u|}^\sigma) = f(x,u) \end{equation*} $
in a smooth bounded domain $ \Omega\subset{\mathbb{R}}^N $ with Dirichlet boundary conditions. Our result covers some typical physical models.
Citation: Rui Sun. Soliton solutions for a class of generalized quasilinear Schrödinger equations[J]. AIMS Mathematics, 2021, 6(9): 9660-9674. doi: 10.3934/math.2021563
In this paper, critical point theory is used to show the existence of nontrivial solutions for a class of generalized quasilinear Schrödinger equations
$ \begin{equation*} -\Delta_pu-{|u|}^{\sigma-2}uh'({|u|}^\sigma)\Delta_ph({|u|}^\sigma) = f(x,u) \end{equation*} $
in a smooth bounded domain $ \Omega\subset{\mathbb{R}}^N $ with Dirichlet boundary conditions. Our result covers some typical physical models.
| [1] |
J. Q. Liu, Y. Q. Wang, Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equations, 187 (2003), 473–493. doi: 10.1016/S0022-0396(02)00064-5
|
| [2] | J. Q. Liu, Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅰ, Proc. Am. Math. Soc., 131 (2003), 441–448. |
| [3] |
M. Poppenberg, K. Schmitt, Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calculus Var. Partial Differ. Equations, 14 (2002), 329–344. doi: 10.1007/s005260100105
|
| [4] |
S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262–3267. doi: 10.1143/JPSJ.50.3262
|
| [5] | A. V. Borovskii, A. L. Galkin, Dynamic modulation of an ultrashort high-intensity laser pulse in matter, JETP, 77 (1993), 562–573. |
| [6] |
B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687–689. doi: 10.1103/PhysRevE.50.R687
|
| [7] |
X. L. Chen, R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082–2085. doi: 10.1103/PhysRevLett.70.2082
|
| [8] |
A. De Bouard, N. Hayashi, J. C. Saut, Global existence of small solotions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73–105. doi: 10.1007/s002200050191
|
| [9] |
A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Japan, 42 (1977), 1824–1835. doi: 10.1143/JPSJ.42.1824
|
| [10] |
E. W. Laedke, K. H. Spatschek, L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764–2769. doi: 10.1063/1.525675
|
| [11] | A. G. Litvak, A. M. Sergeev, One dimensional collapse of plasma waves, J. Exp. Theor. Phys. Lett., 27 (1978), 517–520. |
| [12] |
G. R. W. Quispel, 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
|
| [13] |
H. Lange, B. Toomire, P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys., 36 (1995), 1274–1283. doi: 10.1063/1.531120
|
| [14] |
F. G. Bass, N. N. Nasanov, Nonlinear electromagnetic spin waves, Phys. Rep., 189 (1990), 165–223. doi: 10.1016/0370-1573(90)90093-H
|
| [15] |
R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Physik B, 37 (1980), 83–87. doi: 10.1007/BF01325508
|
| [16] |
V. G. Makhandov, V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep., 104 (1984), 1–86. doi: 10.1016/0370-1573(84)90106-6
|
| [17] | D. C. Liu, Soliton solutions for a quasilinear Schrödinger equation, Electron. J. Differ. Equations, 2013 (2013), 1–13. |
| [18] |
D. C. Liu, P. H. Zhao, Soliton solutions for a quasilinear Schrödinger equation via Morse theory, Proc. Math. Sci., 125 (2015), 307–321. doi: 10.1007/s12044-015-0240-9
|
| [19] |
J. Y. Liu, D. C. Liu, Multiple soliton solutions for a quasilinear Schrödinger equation, Indian J. Pure Appl. Math., 48 (2017), 75–90. doi: 10.1007/s13226-016-0195-2
|
| [20] | L. Zhang, X. H. Tang, Y. Chen, Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation, Topol. Methods Nonlinear Anal., 48 (2016), 539–554. |
| [21] |
L. Zhang, X. H. Tang, Y. Chen, Infinitely many solutions for indefinite quasilinear Schrödinger equations under broken symmetry situations, Math. Methods Appl. Sci., 40 (2017), 979–991. doi: 10.1002/mma.4030
|
| [22] |
L. Zhang, X. H. Tang, Y. Chen, Multiple solutions of sublinear quasilinear Schrödinger equations with small perturbations, Proc. Edinburgh Math. Soc., 62 (2019), 471–488. doi: 10.1017/S0013091518000536
|
| [23] |
Y. T. Shen, Y. J. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal.: Theory Methods Appl., 80 (2013), 194–201. doi: 10.1016/j.na.2012.10.005
|
| [24] |
A. Selvitella, The dual approach to stationary and evolution quasilinear PDEs, Nonlinear Differ. Equations Appl., 23 (2016), 1–22. doi: 10.1007/s00030-016-0354-5
|
| [25] |
M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal.: Theory Methods Appl., 56 (2004), 213–226. doi: 10.1016/j.na.2003.09.008
|
| [26] | P. Lindqvist, On the equation $\text{div}({|\nabla u|}^{p-2}\nabla u)+{\lambda|u|}^{p-2}u = 0$, Proc. Am. Math. Soc., 109 (1990), 157–164. |
| [27] |
X. L. Fan, Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problems, Nonlinear Anal.: Theory Methods Appl., 52 (2003), 1843–1852. doi: 10.1016/S0362-546X(02)00150-5
|
| [28] | M. Willem, Minimax theorems, Boston: Birkhäuser, 1996. |
| [29] |
H. Brézis, L. Nirenberg, Remarks on finding critical points, Commun. Pure Appl. Math., 44 (1991), 939–963. doi: 10.1002/cpa.3160440808
|
Rui Sun. Soliton solutions for a class of generalized quasilinear Schrödinger equations[J]. AIMS Mathematics, 2021, 6(9): 9660-9674. doi: 10.3934/math.2021563
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