Multiplicity of high energy solutions for superlinear Kirchhoff equations

被引：82
作者：
Liu, Wei [1 ]
He, Xiaoming [1 ]
机构：
[1] College of Science, Minzu University of China
来源：
Journal of Applied Mathematics and Computing | 2012年 / 39卷 / 1-2期
基金：
中国国家自然科学基金;
关键词：
High energy solutions; Nonlinear Kirchhoff equations; Variational methods;
D O I：
10.1007/s12190-012-0536-1
中图分类号：
TP [自动化技术、计算机技术];
学科分类号：
0812 ;
摘要：
In this paper, we study the existence of infinitely many high energy solutions for the nonlinear Kirchhoff equations {-(a+b∫ R3 |nabla u| 2dx)Δ u + V(x)u=f(x,u), x∈ℝ 3, u ∈ H 1 (ℝ 3). where a,b>0 are constants, V:ℝ 3→ ℝ is continuous and has a positive infimum. f is a subcritical nonlinearity which needs not to satisfy the usual Ambrosetti-Rabinowitz-type growth conditions. © 2012 Korean Society for Computational and Applied Mathematics.
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收藏
页码：473 / 487
页数：14
相关论文
共 28 条
[1]  
Kirchhoff G., Mechanik, (1883)
[2]  
Bernstein S., Sur une class d'équations fonctionnelles aux dérivé es partielles, Bull. Acad. Sci. URSS, Ser. Math., Izv. Akad. Nauk SSSR, 4, pp. 17-26, (1940)
[3]  
Pohozaev S.I., A certain class of quasilinear hyperbolic equations, Mat. Sb., 96, pp. 152-168, (1975)
[4]  
Lions J.L., On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations. Proc. Internat. Sympos. Inst. Mat. Univ. Fed. Rio de Janeiro, 1997. North-Holland Math. Stud., 30, pp. 284-346, (1978)
[5]  
Arosio A., Panizzi S., On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc., 348, pp. 305-330, (1996)
[6]  
Cavalcanti M.M., Domingos Cavalcanti V.N., Soriano J.A., Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differ. Equ., 6, pp. 701-730, (2001)
[7]  
Chipot M., Lovat B., Some remarks on non local elliptic and parabolic problems, Nonlinear Analysis, Theory, Methods and Applications, 30, 7, pp. 4619-4627, (1997)
[8]  
D'Ancona A.P., Spagnolo S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108, pp. 247-262, (1992)
[9]  
Alves C.O., Correa F.J.S.A., Ma T.F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Computers and Mathematics with Applications, 49, 1, pp. 85-93, (2005)
[10]  
Ma T.F., Munoz Rivera J.E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16, pp. 243-248, (2003)
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