Infinitely many periodic solutions for a semilinear wave equation in a ball in \(\mathbb R^n\) (Q2438824)

The paper deals with the problem \[ \begin{cases} u_{tt}-\Delta u=\mu u+| u| ^{p-1}u,\;\;& t\in \mathbb{R},\;x\in B_R,\\ u(t,x)=0,\;\;& t\in \mathbb{R},\;x\in \partial B_R,\\ u(t+T,x)=u(t,x),\;\;& t\in \mathbb{R},\;x\in B_R, \end{cases} \] where \(0<p<1\), \(B_R=\{x\in \mathbb{R}^n,| x| <R\}\) and \(8R/T=a/b\) with \(a,b\) being relatively prime positive integers, i.e., \((a,b)=1\). The following main theorem is proved: Assume that \(\mu \) is in the resolvent set of the wave operator \(\partial ^2_t-\Delta \) and that \(\mu >\lambda _0\equiv -(n-3)(n-1)/4R^2\) when \(n\equiv 3\pmod {(4,a)}\). Then the above problem has infinitely many radially symmetric weak solutions \(\{u_l\}\) satisfying \[ \int\int _{[0,T]\times [0,R]}| u_l| ^{p+1}r^{n-1}\, \text{d}t\,\text{d}r\rightarrow 0 \;\text{ as } l\rightarrow \infty . \]