EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

doi:10.1016/S0252-9602(17)30113-3

Abstract

Abstract:

In this paper, we study the following generalized quasilinear Schrödinger equations with critical or supercritical growths
-div(g²(u)∇u) + g(u)g'(u)|∇u|2 + V (x)u=f(x, u) + λ|u|^p-2u, x ∈ R^N,
where λ > 0, N ≥ 3, g:R → R⁺ is a C¹ even function, g(0)=1, g'(s) ≥ 0 for all s ≥ 0,???20170623???((g(s))/(|s|^α-1)):=β > 0 for some α ≥ 1 and (α-1)g(s) > g'(s)s for all s > 0 and p ≥ α2^*. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 using a change of variables and variational method.

Key words: quasilinear Schrödinger equations, critical or supercritical growths, variational methods

Quanqing LI, Xian WU. EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS[J].Acta mathematica scientia,Series B, 2017, 37(6): 1870-1880.

Trendmd

References

[1] Brüll L, Lange H. Solitary waves for quasilinear Schrödinger equations. Expo Math, 1986, 4:278-288
[2] Brandi H, Manus C, Mainfry G, Lehner T, Bonnaud G. Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys Fluids B, 1993, 5:3539-3550
[3] Bass F G, Nasanov N N. Nonlinear electromagnetic-spin waves. Phys Rep, 1990, 189:165-223
[4] Cuccagna S. On instability of excited states of the nonlinear Schrödinger equation. Phys D, 2009, 238:38-54
[5] Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation:a dual approach. Nonlinear Anal, 2004, 56:213-226
[6] Chen X L, Sudan R N. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys Rev Lett, 1993, 70:2082-2085
[7] De Bouard A, Hayashi N, Saut J. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm Math Phys, 1997, 189:73-105
[8] Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differ Equ, 2015, 258:115-147
[9] Deng Y, Peng S, Yan S. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Differ Equ, 2016, 260:1228-1262
[10] Makhankov V G, Fedyanin V K. Nonlinear effects in quasi-one-dimensional models and condensed matter theory. Phys Rep, 1984, 104:1-86
[11] Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys B, 1980, 37:83-87
[12] Fleming W H. A selection-migration model in population genetic. J Math Biol, 1975, 20:219-233
[13] Kurihara S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Jpn, 1981, 50:3262-3267
[14] Kelley P L. Self focusing of optical beams. Phys Rev Lett, 1965, 15:1005-1008
[15] Lange H, Poppenberg M, Teismann H. Nash-Moser methods for the solution of quasilinear Schrödinger equations. Comm Partial Differ Equ, 1999, 24:1399-1418
[16] Laedke E, Spatschek K, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1983, 24:2764-2769
[17] Liu J, Wang Z. Soliton solutions for quasilinear Schrödinger equations, I. Proc Amer Math Soc, 2002, 131:441-448
[18] Liu J, Wang Y, Wang Z. Soliton solutions for quasilinear Schrödinger equations, Ⅱ. J Differ Equ, 2003, 187:473-493
[19] Liu J, Wang Y, Wang Z. Solutions for quasilinear Schrödinger equations via the Nehari Method. Comm Partial Differ Equ, 2004, 29:879-901
[20] Moameni A. Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in R^N. J Differ Equ, 2006, 229:570-587
[21] Kosevich A M, Ivanov B A, Kovalev A S. Magnetic solitons. Phys Rep, 1990, 194:117-238
[22] Willem M. Minimax Theorem. Boston, MA:Birkhäuser Boston, Inc, 1996
[23] Poppenberg M, Schmitt K, Wang Z. On the existence of soliton solutons to quasilinear Schrödinger equations. Calc Var Partial Differ Equ, 2002, 14:329-344
[24] Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interaction. Phys Rev E, 1994, 50:687-689
[25] Quispel G R W, Capel H W. Equation of motion for the Heisenberg spin chain. Phys A, 1982, 110:41-80
[26] Silva E A B, Vieira G F. Quasilinear asymptotically periodic Schrödinger equations with critical growth. Calc Var, 2010, 39:1-33
[27] Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal, 2013, 80:194-201
[28] Wang Y. A class of quasilinear Schrödinger equations with critical or supercritical exponents. Comput Math Appl, 2015, 70:562-572
[29] Wu X. Multiple solutions for quasilinear Schrödinger equations with a parameter. J Differ Equ, 2014, 256:2619-2632