# Bifurcation points of a degenerate elliptic boundary-value problem

### Charles A. Stuart

Ecole Polytechnique Federale, Lausanne, Switzerland### Gilles Evéquoz

Ecole Polytechnique Federale, Lausanne, Switzerland

## Abstract

We consider the nonlinear elliptic eigenvalue problem \begin{align*} -\nabla\cdot\{A(x)\nabla u(x)\} & =\lambda f(u(x))\text{ for }x\in\Omega\\ u(x) & =0\text{ for }x\in\partial\Omega \end{align*} where $Ω$ is a bounded open subset of $R_{N}$ and $f∈C_{1}(R)$ with $f(0)=0$ and $f_{′}(0)=1$. The ellipticity is degenerate in the sense that $0∈Ω$ and $A(x)>0$ for $x=0$ but $lim_{x→0}∣x∣_{2}A(x) =1$. We show that there is vertical bifurcation at all points $λ$ in the interval $(4N_{2} ,∞).$ Bifurcation also occurs at any eigenvalues of the linearized problem that are below $4N_{2} $. Our treatment is based on recent results concerning the bifurcation points of equations with nonlinearities that are Hadamard differentiable, but not Fr\'{e}chet differentiable.

## Cite this article

Charles A. Stuart, Gilles Evéquoz, Bifurcation points of a degenerate elliptic boundary-value problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 4, pp. 309–334

DOI 10.4171/RLM/471