On the existence of periodic solutions for semilinear wave equation on a ball in n with the space dimension n odd

Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball

“…Moreover, it follows from the methods of this paper that there exists an accumulation point of the spectrum on the interval [−2πq n , −q n ]. Similar results hold when α is an arbitrary rational number [3].…”

“…It has been proved in [3] that if α = 1/4, n > 3 is odd and if we set q n = 1 π 2 (n − 1)(n − 3), then every element of the spectrum outside of [−2πq n , −q n ] is an isolated eigenvalue with finite multiplicity, and that 0 is not in the spectrum. Moreover, it follows from the methods of this paper that there exists an accumulation point of the spectrum on the interval [−2πq n , −q n ].…”

“…, when α belongs to the same class of irrationals, and to combine the number theory techniques used in [4] and the asymptotic properties of the Bessel functions used in [5] and [3] to show in Theorem 1 of Section 3 that, for sufficiently large α with bounded partial quotients, 0 is not an accumulation point of the spectrum.…”

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

“…Thus the operator does not have a compact inverse on the range of , and the loss of compactness may cause some difficulties to solve the problem (1.1). Combining the fixed point theory with a reduction argument, Ben-Naoum and Berkovits [5] proved the existence and uniqueness result of problem (1.1) under some extra symmetrical assumptions on the nonlinear term. The result in [5] based upon the property that 0 is not in the spectral set of the n-dimensional wave operator when R = π/2 and T = 2π.…”

“…For examples one may refer to [5][6][7]13,14,[20][21][22]. In contrast to the one-dimensional problem, the spectral properties of the multi-dimensional problem in the radially symmetric case are more difficult to study as they rely on the asymptotic behaviors of the Bessel functions.…”