Resumen de A bifurcation problem governed by the boundary condition II

Jorge José García Melián , Julio Daniel Rossi , José Claudio Sabina de Lis

• In this work we consider the problem $\Delta u = a(x) u^p$ in $\Omega$, $\frac{\partial u}{\partial \nu} = \lambda u$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain, $\nu$ is the outward unit normal to $\partial\Omega$, $\lambda$ is regarded as a parameter, and $0 < p < 1$. We consider both cases where $a(x) > 0$ in $\Omega$ or $a(x)$ is allowed to vanish in a whole subdomain $\Omega_0$ of $\Omega$. Our main results include existence of non-negative non-trivial solutions in the range $0 < \lambda < \sigma_1$, where $\sigma_1$ is characterized by means of an eigenvalue problem, uniqueness and bifurcation from infinity of such solutions for small $\lambda$, and the appearance of dead cores for large enough $\lambda$