Whispering Gallery Modes for Semilinear Dirichlet Eigenvalue Problems

We study the boundary localization phenomenon, known as whispering gallery modes, for weak solutions to semilinear Dirichlet eigenvalue problems in the unit ball $B_1 \subseteq \mathbb{R}^d$ ($d \geq 2$) of the form $$\begin{cases} -\Delta u + f(u) = \lambda u & \text{in } B_1,\\ u = 0 & \text{on } \partial B_1. \end{cases}$$ Here, $f = F'$ where $F$ is a nonnegative $C^2$-function with superquadratic polynomial growth. We prove the existence of a sequence of solutions $(u_n, \lambda_n)$ with $\lambda_n \to +\infty$ such that, for any $\tau \in (0,1)$, $$\lim_{n \to \infty} \frac{E_\tau(u_n)}{E_1(u_n)} = 0,$$ where $E_\rho(u) = \int_{B_\rho} \bigl( \frac{1}{2} |\nabla u|^2 + F(u) \bigr) \, dx$ is the energy over the ball of radius $\rho$. This establishes that the energy of these high-eigenvalue solutions concentrates near the boundary, extending the classical whispering gallery mode phenomenon from linear Laplacian eigenfunctions to the semilinear setting. As a direct application, the case $F(u) = u^4 /4$ yields, after suitable scaling, a sequence of high-eigenvalue boundary-concentrating solutions, providing a nonlinear analogue of whispering gallery modes for the Allen-Cahn equations. The approach combines spectral properties of the linear Laplacian with local nonlinear bifurcation and is expected to adapt to related interior localization phenomena, such as bouncing ball modes, in other geometries where linear eigenfunctions exhibit analogous localization behaviors (e.g., filled ellipses or elliptical annuli).