Acta mathematica scientia, Series B >
NONSMOOTH CRITICAL POINT THEOREMS AND ITS APPLICATIONS TO QUASILINEAR SCHRÖDINGER EQUATIONS
Received date: 2014-12-05
Revised date: 2015-05-13
Online published: 2016-01-30
Supported by
The first author was supported by NSF of China(11201488) and Hunan Provincial Natural Science Foundation of China(14JJ4002). The second author was supported by NSF of China(11371146).
In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear Schrödinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter.
Zhouxin LI , Yaotian SHEN . NONSMOOTH CRITICAL POINT THEOREMS AND ITS APPLICATIONS TO QUASILINEAR SCHRÖDINGER EQUATIONS[J]. Acta mathematica scientia, Series B, 2016 , 36(1) : 73 -86 . DOI: 10.1016/S0252-9602(15)30079-5
[1] Borovskii A V, Galkin A L. Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP, 1993, 77:562-573
[2] Aouaoui S. Multiplicity of solutions for quasilinear elliptic equations in RN. J Math Anal Appl, 2010, 370:639-648
[3] Bartsch T, Pankov A, Wang Z Q. Nonlinear Schrödinger equations with steep potential well. Comm Contemp Math, 2001, 4:549-569
[4] do O J M, Severo U. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Comm Pure Appl Anal, 2009, 8:621-644
[5] Canino A, Degiovanni M. Nonsmooth critical point theory and quasilinear elliptic equations//Granas A, Frigon M, Sabidussi G, eds. Topological Methods in Differential Equations and Inclusions, Montreal, 1994, NATO ASI Series. Dordrecht:Kluwer Academic Publishers, 1995:1-50
[6] Corvellec J N, Degiovanni M, Marzocchi M. Deformation properties for continuous functionals and critical point theory. Topol Methods Nonl Anal, 1993, 1:151-171
[7] Degiovanni M, Marzocchi M. A critical point theory for nonsmooth functional. Ann Mat Pura Appl, 1994, 167(4):73-100
[8] Arcoya D, Boccardo L. Critical points for multiple integrals of the calculus of variations. Arch Rational Meth Anal, 1996, 134:249-274
[9] Arcoya D, Boccardo L, Orsina L. Existence of critical points for some noncoercive functionals. Ann I H Poincaré-AN, 2001, 18(4):437-457
[10] Arioli G, Gazzola F. Quasilinear elliptic equations at critical growth. NoDEA Nonl Eiffer Equ Appl, 1998, 5:83-97
[11] Canino A. Multiplicity of solutions for quasilinear elliptic equations. Topol Methods Nonl Anal, 1995, 6:357-370
[12] Shen Y. Nontrivial solution for a class of quasilinear equation with natural growth. Acta Math Sinica, Chinese Series, 2003, 46:683-690
[13] Li Z, Shen Y, Yao Y. Nontrivial solutions for quasilinear elliptic equations with natural growth. Acta Math Sinica, Chinese Series, 2009, 52:785-798
[14] Shen Y, Li Z, Wang Y. Sign-changing critical points for noncoercive functionals. Topol Methods Nonl Anal, 2014, 43:373-384
[15] Squassina M. Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems. Electronic Journal of Differential Equations, Monograph 7, 2006
[16] Abdellaoui B, Boccardo L, Peral I, et al. Quasilinear elliptic equations with natural growth. Differ Integr Equ, 2007, 20:1005-1020
[17] Boccardo L, Spagnolo S. Positive solutions for some quasilinear elliptic equations with natural growths. Rend Mat Acc Lincei, 2000, 11:31-39
[18] Li Z. Existence of nontrivial solutions for quasilinear elliptic equations at critical growth. Appl Math Comput, 2011, 218:76-87
[19] Pellacci B, Squassina M. Unbounded critical points for a class of lower semicontinuous functionals. J Differ Equ, 2004, 201:25-62
[20] Arioli G, Gazzola F. On a quasilinear elliptic differential equation in unbounded domains. Rend Istit Mat Univ, Trieste, 1998, 30:113-128
[21] Conti M, Gazzola F. Positive entire solutions of quasilinear elliptic problems via nonsmooth critical point theory. Topol Methods Nonl Anal, 1996, 8:275-294
[22] Gazzola F. Positive solutions of critical quasilinear elliptic problems in general domains. Abstr Appl Anal, 1998, 3(1/2):65-84
[23] Liu J Q, Wang Z Q, Guo Y X. Multibump solutions for quasilinear elliptic equations. J Funct Anal, 2012, 262:4040-4102
[24] Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schródinger. Nonl Anal TMA, 2013, 80:194-201
[25] Li Z, Shen Y, Zhang Y. An application of nonsmooth critical point theory. Topol Methods Nonl Anal, 2010, 35:203-219
[26] Boccardo L, Murat F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonl Anal TMA, 1992, 19:581-597
[27] Pucci P, Serrin J. A general variational identity. Indiana Univ Math J, 1986, 35(3):681-703
[28] Degiovanni M, Musesti A, Squassina M. On the regularity of solutions in the Pucci-Serrin identity. Calc Var Partial Differential Equations, 2003, 18:317-334
/
| 〈 |
|
〉 |