JournalsrlmVol. 17, No. 4pp. 309–334

Bifurcation points of a degenerate elliptic boundary-value problem

  • Charles A. Stuart

    Ecole Polytechnique Federale, Lausanne, Switzerland
  • Gilles Evéquoz

    Ecole Polytechnique Federale, Lausanne, Switzerland
Bifurcation points of a degenerate elliptic boundary-value problem cover
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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\Omega is a bounded open subset of RN\mathbb{R}^{N} and fC1(R)f\in C^{1}(\mathbb{R}) with f(0)=0f(0)=0 and f(0)=1f^{\prime}(0)=1. The ellipticity is degenerate in the sense that 0Ω0\in\Omega and A(x)>0A(x)>0 for x0x\neq0 but limx0A(x)x2=1\lim_{x\rightarrow0}\frac{A(x)}{\left\vert x\right\vert ^{2}}=1. We show that there is vertical bifurcation at all points λ\lambda in the interval (N24,).(\frac{N^{2}}{4},\infty). Bifurcation also occurs at any eigenvalues of the linearized problem that are below N24\frac{N^{2}}{4}. 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