| Abstract: |
| We deal with standing waves for the wave equation with hyperbolic boundary conditions, posed in a bounded domain with regular boundary. These problems possess a wide literature, including the papers on Arch. Rat. Mech. Anal. (2017), J.D.E. (2018) and DCDS-S (2021) by the author.
Stationary solutions of these evolution problems turn out to be solutions of a doubly elliptic problem posed in the domain. This problem involved the Laplace operator inside the domain, the Laplace--Beltrami operator at the boundary, and up to two nonlinear sources, one inside the domain, the other one at the boundary.
This type of problem has been studied by the author in a paper on Comm. Anal. Mech (2023) in presence of a single homogeneous boundary source.
In this talk we discuss the more involved case of two nonlinear, possibly vanishing and nonhomogeneous, sources, which can also involve linear terms. In particular, we shall characterize in several ways the potential--well depth, by also proving the existence of solutions at this level. Finally, we shall also give multiplicity results for stationary solutions. |
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