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

Abstract

This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation (t, x, u) 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.

Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an n-Dimensional Ball

“…It is known (see, for example, [3,8]) that L 0 is a symmetric operator on L 2 (Ω, ρ), and the spectrum of the linear operator L 0 is made of eigenvalues…”

“…In the past few years, many authors (see [6,7,11,12,13,14,15,16,17,18]) payed much attention to the periodic solutions of the wave equation when the space dimension n = 1. There are also many papers (see [2,3,8,9,10,16]) consider the wave equation in a ball with radius R and time period T , but a very interest thing is that the solvability of wave equation in an n-dimensional ball with radius R depends on the arithmetical properties of R and T . It is well known that, for the one-dimensional case, if T = 2π and radius R = π/2, the structure of the spectral set of the wave operator is made of the eigenvalues λ jk = (2j − 1) 2 − k 2 (j ∈ Z + = {1, 2, · · · }, k ∈ Z).…”

“…Later, they [9] also dealt with the wave equation u tt − ∆u = µu + a(t, x)|u| p−1 u in a ball with R = π/2 and obtained infinitely many 2π-periodic solutions for the cases that n is an even integer or n > 3 is odd. Recently, they [8] also investigated the wave equation u tt −∆u = g(t, x, u) and proved that there exists at least three radially symmetric periodic solutions under some certain suitable conditions for the case that n = 2 or n > 3 is odd and R/T = d/4, d ∈ Z + . In this case, it is proved that 0 is not in the spectral set of the wave operator, which is a crucial fact used in the work.…”

Dynamics of nonlinear hyperbolic equations of Kirchhoff type

“…Many mathematicians have devoted their effort to improve Sattinger's result; see the papers of Cazenave [8], Gazzola and Squassina [17], Liu [29], Liu and Zhao [30] which solve the initial-boundary value problem (1.3), and the papers of Brézis [6], Chen and Zhang [11]- [13], Ding et al [15], Rabinowitz [40], Schechter [42] which concern the periodic solutions for (1.3). When a > 0, the problem (1.1) is nonlocal due to the presence of the integro-differential term Ω |∇u| 2 dx ∆u, which arises in many interesting models in physics, biology as well as other areas.…”

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

“…Thereafter, many authors, such as Brézis, Chang, Nirenberg etc., have used and developed the variational methods, topological degree and index theory to obtain a lot of results on the existence and multiplicity of periodic solutions for the problem with various nonlinearities (see e.g. [1,3,5,9,11,13,14,15,16,17,25,26,30] and the references therein).…”