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Session 38. Variational Methods in Nonlinear Analysis
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A dual variational approach to nonlinear Helmholtz equations
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Tobias Weth, Institut of Mathematics, Goethe University Frankfurt a.M., Germany
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The talk is based on the joint work with Gilles Evequoz (Frankfurt)
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We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation
\[
-\Delta u-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}({\mathbb{R}}^N)
\]
with \(N\geq 3\), \(\frac{2(N+1)}{(N-1)}< p<\frac{2N}{N-2}\) and
nonnegative \(Q \in
L^\infty({\mathbb R}^N)\). We prove the existence of nontrivial solutions for
periodic \(Q\) as well as in the case where \(Q(x)\to 0\) as
\(|x|\to\infty\). Classical direct methods in critical point theory do not apply to this problem due to the lack of Fredholm properties. In the periodic case, a key ingredient of the approach
is a new nonvanishing
theorem related to an associated integral equation. The solutions we
study are superpositions of outgoing and incoming waves and are
characterized by a nonlinear far field relation.
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