Hadamard differentiability and bifurcation

Abstract

In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form

$$ -\Delta u(x)+q(x)u(x)=\lambda\eta(x)^{-1}f(\eta(x)u(x)) $$

and

$$ -\Delta u(x)+q(x)u(x)-\eta(x)^{-1}f(\eta(x)u(x))=\lambda u(x), $$

where $u\in H^{2}(\mathbb{R}^{N})$.