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

TL;DR

This paper studies the existence and multiplicity of solutions for a Robin boundary value problem with sign-changing nonlinearities, showing how solutions behave as boundary parameters vary.

Contribution

It establishes the existence of multiple solutions for the Robin problem with sign-changing nonlinearities and analyzes their limiting behavior as boundary conditions change.

Findings

Existence of two nonnegative solutions for large mbda.

Solutions' maximum lies between two zeros of the nonlinearity.

Solution set degenerates to Neumann or Dirichlet solutions as mma varies.

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 $\gamma\to\infty$.