Instability of nonnegative solutions for a class of semipositone problems

Abstract:

We consider the boundary value problem \[ \begin {gathered} - \Delta u(x) = \lambda f(u(x)),\quad x \in \Omega \hfill \\ Bu(x) = 0,\quad x \in \partial \Omega \hfill \\ \end {gathered} \] where $\Omega$ is a bounded region in ${R^N}$ with smooth boundary, $Bu = \alpha h(x)u + (1 - \alpha )\partial u/\partial n$ where $\alpha \in [0,1]h:\partial \Omega \to {R^ + }$ with $h = 1$ when $\alpha = 1$, $\lambda > 0,f$ is a smooth function with $f(0) < 0$ (semipositone), $f’(u) > 0$ for $u > 0$ and $f''(u) \geq 0$ for $u > 0$. We prove that every nonnegative solution is unstable.