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 is a bounded open subset of and with and . The ellipticity is degenerate in the sense that and for but . We show that there is vertical bifurcation at all points in the interval Bifurcation also occurs at any eigenvalues of the linearized problem that are below . 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