On the bifurcation theory of semilinear elliptic eigenvalue problems

Abstract:

A standard “bootstrap” method is used to show that the bifurcation problem for the semilinear eigenvalue problem $\Delta u + \lambda f(x,u) = 0$ in $\Omega$, $u{|_{\partial \Omega }} = 0$, where $f(x,0) \equiv 0$, and $(\partial /\partial u)f(x,0) > 0$, and when formulated in terms of weak solutions, is a local problem, i.e. independent of the behavior of $f$ for large $u$. A principle of linearization for this problem is proved under mild differentiability conditions on $f$.