On the spectrum of a mixed-type operator with applications to rotating wave solutions (Q2680539)

The author investigates the existence of rotating wave solutions of a nonlinear wave equation which reduces, via the ansatz \(v(t,x)=u(R_{\alpha t}(x))\) with the planar rotation operator \(R_{\theta}\), to the consideration of the problem \[ \begin{cases} -\Delta u + \alpha^2 \partial_\theta^2 u + mu = |u|^{p-2} u & \text{ in }B,\\ u=0 &\text{ on }\partial B, \end{cases} \] where \(\alpha>0\), \(\partial_\theta = x_1 \partial_{x_2} - x_2 \partial_{x_1}\) corresponds to the angular derivative, \(m>0\), \(p \in (2,4)\), and \(B\) is the unit disk in \(\mathbb{R}^2\). The case \(\alpha \leq 1\) has been studied in [\textit{J. Kübler} and \textit{T. Weth}, ``Rotating waves in nonlinear media and critical degenerate Sobolev inequalities'', Preprint, \url{arXiv:2203.07991}], so the author is concentrated on the case \(\alpha>1\). The Dirichlet eigenvalues of the operator \(-\Delta + \alpha^2 \partial_\theta^2\) in \(B\) are given by \(j_{l,k}^2 - \alpha^2 l^2\), for \(l \in \mathbb{N}_0\), \(k \in \mathbb{N}\), and \(j_{l,k}\) stands for the \(k\)-th positive zero of the Bessel function \(J_l\). That is, properties of the spectrum are connected with properties of the Bessel zeros. As a part of main results, the author proves the existence of an unbounded sequence \(\{\alpha_n\} \subset (1,+\infty)\) such that, for any \(n\), the spectrum of \(-\Delta + \alpha_n^2 \partial_\theta^2\) has no accumulation points and each eigenvalue is of finite multiplicity. To this end, an asymptotic behavior of \(j_{xk,k}/k\) is investigated, and it improves the one obtained in [\textit{Á. Elbert} and \textit{A. Laforgia}, J. Math. Anal. Appl. 98, 502--511 (1984; Zbl 0549.33007)]. Then, by considering a minimization problem for the strongly indefinite energy functional over the Nehari-Pankov manifold, the author proves the existence of a ground state of the problem for any \(n\). Moreover, if \(m\) is large enough, then the ground state is nonradial. The results for complex-valued solutions of the initial nonlinear wave equation are also discussed.