Annali di Matematica Pura ed Applicata

, Volume 119, Issue 1, pp 379–390

# Existence et determination du premier point de bifurcation pour des couples d'opérateurs potentiels non différentiables à l'origine

• Hugo Beirão da Viega
Article

## Summary

Let Λ and Γ denote respectively the gradients of two Gateaux differentiable real functionals ϕ and ϕ on a real reflexive Banach space V. We shall prove that, under very general assumptions, the first bifurcation point (at the origin in V) for the non-linear equation Γ(u)=λΛ(u) is the first eigenvalue of the linear equation B(u)=λA(u) where Λ=A+ν and Γ=B+ω. The perators Λ and Γ are not necessarily differentiable at the origin. Roughly speaking it sufficies to assume that the projections in a suitable direction of the remainders ν(u) and ω(u) go to zero faster than ∥u∥; this direction depends on u ans is, in the simplest examples, actually the u-direction.

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