Bifurcation points of a degenerate elliptic boundary-value problem

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 $\Omega$ 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 $x\neq0$ 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 Fr\'{e}chet differentiable.