Gilles Evéquoz

Hadamard differentiability and bifurcation

Keywords: conditions for bifurcation Reference ANA-ARTICLE-2007-001View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

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−k2u=Q(x)|u|p−2u,u∈W2,p(RN) with N≥3, 2(N+1)(N−1)<p<2NN−2 and nonnegative Q∈L∞(RN). We prove the existence of nontrivial solutions for periodic Q as well as in the case where Q(x)→0 as |x|→∞. 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.

Real Solutions to the Nonlinear Helmholtz Equation with Local Nonlinearity

AbstractIn this paper, we study real solutions of the nonlinear Helmholtz equation

Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part

- \Delta u - k^2 u = f(x,u),\quad x\in \mathbb{R}^N

Bifurcation and concentration of radial solutions of a nonlinear degenerate elliptic eigenvalue problem

-Δu-k2u=f(x,u),x∈RNsatisfying the asymptotic conditions

Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation

u(x)=O\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm and} \quad \frac{\partial^2 u}{\partial r^2}(x)+k^2u(x)=o\left(|x|^{\frac{1-N}{2}}\right) \quad {\rm as}\, r=|x| \to\infty.