In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{ equation*} \begin{ cases} -\Delta u = \lambda f(u) & \text{ in } \Omega, \frac{ \partial u}{ \partial \nu} + \gamma u=0 & \text{ on } \partial\Omega, \end{ cases} \end{ equation*} where $\Omega\subset \mathbb{ R}^N$ ($N\geq 1$) is a smooth bounded domain, and $\lambda, \gamma>0$. Our main assumption is that $f\colon \mathbb{ R}\to \mathbb{ R}$ is a locally Lipschitz function, possibly sign-changing, such that $f(s)>0$ for every $s\in (\alpha,\beta)$, where $0<\alpha<\beta$ are two zeros of $f$. Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in $(\alpha,\beta)$ for sufficiently large $\lambda$. Moreover, we analyse the limiting behaviour of the solution set of this Robin problem, showing that it degenerates into that of the associated Neumann problem as $\gamma\to 0$ and into that of the associated Dirichlet problem as $\gamma\to\infty$.
Categories: math.AP