JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (6): 81-87.doi: 10.6040/j.issn.1671-9352.0.2018.620

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Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms

ZHANG Nian, JIA Gao*   

  1. School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
  • Published:2019-06-05

Abstract: We study a class of fourth-order elliptic equations with nonlocal term,{Δ2u-(a+b∫RN|∇u|2dx)Δu+V(x)u-1/2Δ(u2)u=f(x,u), x∈RN,u(x)∈H 2(RN),Where N≤5, constants a>0, b≥0, Δ2=Δ(Δ)is the biharmonic operater, the nonlinearity f(x,u) doesnt satisfy AR condition and the potential function V(x) is also allowed to be sign-changing. We establish the existence of a sequence of high energy weak solutions for this class of elliptic equations via variational methods.

Key words: Kirchhoff type equation, sign-changing potential, variational methods, Fountain theorem, infinitely many solutions

CLC Number: 

  • O175
[1] LAZERA C, MCKENNA. P J. Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis[J]. SIAM Review, 1990, 32(4):537-578.
[2] SUN Juntao, WUT Sungfang. Ground state solutions for an indefinite Kirchhoff type problem with steep potential well[J]. Journal of Differential Equations, 2014, 256(4):1771-1792.
[3] FANG Xiangdong, HAN Zhiqing. Existence of a ground state solution for a quasilinear Schrödinger equation[J]. Advance Nonlinear Studies, 2014, 14(4):941-950.
[4] CHIPOT M, LOVAT B. Some remarks on nonlocal elliptic and parabolic problems[J]. Nonlinear Analysis Theory Methods & Applications, 1997, 30(7):4619-4627.
[5] MAO Anmin, CHANG Hejie. Kirchhoff type problems in RN with radial potentials and locally Lipschitz functional[J]. Applied Mathematics Letters, 2016, 62:49-54.
[6] LI Hongying. Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential [J]. Computers & Mathematics with Applications, 2018, 85(8):2858-2873.
[7] LI Yuhua, LI Fuyi, SHI Junping. Existence of a positive solution to Kirchhoff type problems without compactness conditions [J]. Journal of Differential Equations, 2012, 253(7): 2285-2294.
[8] ZHANG Jian, TANG Xianhua, ZHANG Wen. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential[J]. Applied Mathematics & Computation, 2014, 242:491-499.
[9] WU Xian. Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in RN[J]. Nonlinear Analysis, 2012, 75(8):3470-3479.
[10] CHENG Biao, TANG Xianhua. High energy solutions of modified quasilinear fourth-order elliptic equations with sigh-changing potential[J]. Computers & Mathematics with Applications, 2017, 73(1):27-36.
[11] CHEN Shaoxiong, LIU Jiu, WU Xian. Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on RN[J]. Applied Mathematics & Computation, 2014, 248:593-601.
[12] SALVATORE A. Multiple solutions for perturbed elliptic equations in unbounded domains[J]. Advanced Nonlinear Studies, 2003, 3(1):1-23.
[13] RABINOWITZ P H. Minimax methods in critical point theory with applications to differential equations[M]. Rhode Island: American Mathematical Society, 1986.
[14] WILLEM M. Minimax theorems[M]. Boston: Birkhäuser, 1996.
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