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Existence of multiple periodic solutions to asymptotically linear wave equations in a ball

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Abstract

This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation

$$\begin{aligned} u_{tt}-\Delta u = g(t,x,u) \end{aligned}$$

in a n-dimensional ball with radius R, where \(n>1\) and g(t, x, u) is radially symmetric in x and T-periodic in time. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of R and T. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to construct at least three radially symmetric solutions with time period T, when T is a rational multiple of R and g(t, x, u) satisfies some monotonicity and asymptotically linear conditions.

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Authors and Affiliations

  1. Science and Information College, Qingdao Agricultural University, Qingdao, 266109, People’s Republic of China

    Jianyi Chen

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Zhitao Zhang

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Zhitao Zhang

Authors
  1. Jianyi Chen
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  2. Zhitao Zhang
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Correspondence to Zhitao Zhang.

Additional information

Communicated by P.Rabinowitz.

Supported by National Natural Science Foundation of China (11325107, 11271353, 11331010), Natural Science Foundation of Shandong Province (ZR2016AQ04), Research Foundation for Advanced Talents of Qingdao Agricultural University (No. 6631114328) and the Science and Technology Program of Shandong Universities (J16LI04).

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Chen, J., Zhang, Z. Existence of multiple periodic solutions to asymptotically linear wave equations in a ball. Calc. Var. 56, 58 (2017). https://doi.org/10.1007/s00526-017-1154-4

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  • DOI: https://doi.org/10.1007/s00526-017-1154-4

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