Bifurcation points of a degenerate elliptic boundary-value problem
Gilles Évéquoz, C. A. Stuart · Rendiconti Lincei Matematica e Applicazioni · 2006
We consider the nonlinear elliptic eigenvalue problem \begin{align*} - abla\cdot\{A(x) abla u(x)\} & =\lambda f(u(x))\text{ for }x\in\Omega\\u(x) & =0\text{ for }x\in\partial\Omega \end{align*} where \Omega is a bounded open subset of \mathbb{R}^{N} and f\in C^{1}(\mathbb{R}) with f(0)=0 and f^{\prime}(0)=1 . The ellipticity is degenerate in the sense that 0\in\Omega and A(x)>0 for x eq0 but \lim_{x\rightarrow0}\frac{A(x)}{\left\vert x\right\vert ^{2}}=1 . We show that there is vertical bifurcation at all points \lambda in the interval (\frac{N^{2}}{4},\infty). Bifurcation also occurs at any eigenvalues of the linearized problem that are below \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 Fréchet differentiable.