arXiv subjects

Gilles Evequoz

Publications and source records attributed to Gilles Evequoz.

2 recordsLinked to original sources

arXiv2015

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -Δu-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ 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 $|x|\to\infty$. 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.

math.AP↗

arXiv2013

Real solutions to the nonlinear Helmholtz equation with local nonlinearity

In this paper, we study real solutions of the nonlinear Helmholtz equation $$ - Δu - k^2 u = f(x,u),\qquad x\in \R^N $$ satisfying the asymptotic conditions $$ u(x)=O(|x|^{\frac{1-N}{2}}) \quad \text{and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2 u(x)) =o(|x|^{\frac{1-N}{2}}) \qquad \text{as $r=|x| \to \infty$.} $$ We develop the variational framework to prove the existence of nontrivial solutions for compactly supported nonlinearities without any symmetry assumptions. In addition, we consider the radial case in which, for a larger class of nonlinearities, infinitely many solutions are shown to exist. Our results give rise to the existence of standing wave solutions of corresponding nonlinear Klein-Gordon equations with arbitrarily large frequency.

math.AP↗