带类p(x)-拉普拉斯算子的双非局部问题的无穷多解

数学物理学报 ›› 2018, Vol. 38 ›› Issue (3): 514-526.

带类p(x)-拉普拉斯算子的双非局部问题的无穷多解

张申贵

西北民族大学数学与计算机科学学院兰州 730030

收稿日期:2016-09-14 修回日期:2017-10-16 出版日期:2018-06-26 发布日期:2018-06-26
作者简介:张申贵,E-mail:zhangshengui315@163.com
基金资助:
国家自然科学基金（31260098）、甘肃省科技计划项目（1610RJZA102）和中央高校基本科研业务专项经费（31920170147）

Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator

Zhang Shengui

School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030

Received:2016-09-14 Revised:2017-10-16 Online:2018-06-26 Published:2018-06-26
Supported by:
Supported by the NSFC (31260098), the Science and Technology Planning Project of Gansu Province (1610RJZA102) and the Fundamental Research Funds for the Central Universities (31920170147)

摘要/Abstract

摘要： 该文运用变分方法研究一类带类p（x）-拉普拉斯算子的双非局部狄利克雷问题.利用喷泉定理和对称山路定理，得到了此类问题一列高能量和低能量解的存在性.

关键词: 变分方法, 临界点, 类p(x)-拉普拉斯算子, 非局部问题, 弱解

Abstract: In this article, we study a class of bi-nonlocal Dirichlet problem involving p(x)-Laplacian-like operator via variational methods. Using the fountain theorem and the symmetric mountain pass theorem, the existence of a sequence of high and low energy solutions for this problem are obtained, respectively.

Key words: Variational methods, Critical point, p(x)-Laplacian-like operator, Nonlocal problem, Weak solutions

中图分类号:

O177.91

张申贵. 带类p(x)-拉普拉斯算子的双非局部问题的无穷多解[J]. 数学物理学报, 2018, 38(3): 514-526.

Zhang Shengui. Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator[J]. Acta mathematica scientia,Series A, 2018, 38(3): 514-526.

参考文献

[1] Ruzicka M. Electrorheologial Fluids:Modeling and Mathematial Throry. Berlin:Springer-Verlag, 2000
[2] Zhikov V. Averaging of functionals of the calculus of variations and elasticity theory. Math USSR Izv, 1987, 9:33-66
[3] Chen Y, Levine S, Rao M. Variable exponent, linear growth functionals in image restoration. SIAM J Appl Math, 2006, 66:1383-1406
[4] Avci M, Cekic B, Mashiyev R A. Existence and multiplicity of the solutions of the p(x)-Kirchhoff type equation via genus theory. Mathematical Methods in the Applied Sciences, 2011, 34(14):1751-1759
[5] Allaoui M, Amrouss E, Rachid A. Existence results for a class of nonlocal problems involving p(x)-Laplacian. Mathematical Methods in the Applied Sciences, 2016, 39(4):824-832
[6] Cammaroto F, Vilasi L. Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator. Nonlinear Anal TMA, 2011, 74(5):1841-1852
[7] Chung N T. Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities. Electron J Qual Theory Differential Equations, 2012, 42:1-13
[8] Fan X L, Zhao D. On the spaces L^p(x)(Ω) and W^m,p(x)(Ω). J Math Anal Appl, 2001, 263:424-446
[9] Fan X L, Zhang Q H. Existence of solutions of p(x)-Laplacian Dirichlet problems. Nonlinear Anal TMA, 2003, 52:1843-1852
[10] Fan X L. On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal TMA, 2010, 72:3314-3323
[11] Li L, Tang C L. Existence and multiplicity of solutions for a class of p(x)-biharmonic equations. Acta Mathematica Scientia, 2013, 33B(1):155-170
[12] Massar M, Talbi M, Tsouli N. Multiple solutions for nonlocal system of (p(x), q(x))-Kirchhoff type. Applied Mathematics and Computation, 2014, 242:216-226
[13] Rǎdulescu V D. Nonlinear elliptic equations with variable exponent:old and new. Nonlinear Anal TMA, 2015, 121:336-369
[14] Lapa E C, Rivera V P, Broncano J Q. No-flux boundary problems involving p(x)-Laplacian-like operators. Electronic J Differential Equations, 2015, 219:1-10
[15] Ge B. On superlinear p(x)-Laplacian-like problem without Ambrosetti and Rabinowitz condition. Bull Korean Math Soc, 2014, 51:409-421
[16] Avci M. Ni-Serrin type equations arising from capillarity phenomena with non-standard growth. Boundary Value Problems, 2013, 55(1):1-13
[17] Rodrigues M M. Multiplicity of solutions on a nonlinear eigenvalue problem for p(x)-Laplacian-like operators. Mediterranean J Mathematics, 2012, 9(1):211-223
[18] Zhou Q M. On the superlinear problems involving p(x)-Laplacian-like operators without AR-condition. Nonlinear Anal RWA, 2015, 21:161-169
[19] Ding Y H, Luan S X. Multiple solutions for a class of nonlinear Schrödinger equations. J Differential Equations, 2004, 207:423-457
[20] Zhou H S. An application of a mountain pass theorem. Acta Mathematica Sinica English Series, 2002, 18(1):27-36
[21] Tang C L, Wu X P. Periodic solutions for a class of new superquadratic second order Hamiltonian systems. Appl Math Lett, 2014, 34:65-71
[22] Bartsch T. Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal TMA, 1993, 20:1205-1216
[23] Kajikiya R. A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Analysis, 2005, 225:352-370