Bifurcation points of a degenerate elliptic boundary-value problem

Abstract

We consider the nonlinear elliptic eigenvalue problem

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 $x\ne 0$ but ${\lim }_{x\to 0}\frac{A(x)}{{|x|}^2}=1$. 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.