BZ Seminar in Analysis: Tobias Weth (Frankfurt)

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

Unlike in the case of nonlinear stationary Schršdinger equations, the existence of standing wave solutions to nonlinear Helmholtz equations is relatively unexplored up to now since the linearized problem lacks the Fredholm property in standard Sobolev spaces. I will present a dual variational approach to detect real-valued standing wave solutions in the case of power type nonlinearities of the form $Q(x)|u|^{p-2}u$. Via this approach, existence results will be derived for a periodic weight function $Q$ and in the case where $Q$ decays to zero asymptotically in space. In the periodic case, a key ingredient of the method is a new nonvanishing result related to an associated integral equation. I will also discuss the far field asymptotics of the solutions obtained by this approach. This is joint work with Gilles Evequoz (University of Frankfurt).

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