Multiplicity of solutions for semilinear Robin problems involving sign-changing nonlinearities

Abstract

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 $\gamma\to\infty$.