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

TL;DR

The article analyzes a semilinear Robin boundary value problem with a sign-changing nonlinearity having two zeros $0<\alpha<\beta$, proving the existence of two nonnegative solutions with maxima in $(\alpha,\beta)$ for large $\lambda$ and detailing their asymptotic behavior as the Robin parameter $\gamma$ varies. It introduces an auxiliary truncated problem, employs variational methods and Leray-Schauder degree to obtain multiplicity, and establishes a robust stability framework showing convergence to Neumann limits as $\gamma\to0$ and to Dirichlet limits as $\gamma\to\infty$ (including strong $C(\overline{\Omega})$ convergence). The work shows that Robin solutions serve as an interpolation between Neumann and Dirichlet solutions under varying boundary conditions, and provides precise descriptions of how the solution set degenerates or converges in the two limiting regimes. By combining truncation, degree theory, and Stampacchia-type regularity, the paper delivers new multiplicity and asymptotic results for Robin problems with zeros in the nonlinearity, without requiring Hess-type area conditions for existence.

Abstract

In this article, we investigate the existence and multiplicity of solutions to the Robin problem \begin{equation*} \begin{cases} -Δu = λf(u) & \text{in } Ω, \frac{\partial u}{\partial ν} + γu=0 & \text{on } \partialΩ, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$ ($N\geq 1$) is a smooth bounded domain, and $λ, γ>0$. 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 $s\in (α,β)$, where $0<α<β$ are two zeros of $f$. Without any further conditions, we establish the existence of two nonnegative solutions whose maximum lies in $(α,β)$ for sufficiently large $λ$. 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 $γ\to 0$ and into that of the associated Dirichlet problem as $γ\to\infty$.

Figures (2)

• Figure 1: Sketch of the bifurcation diagram of problem \ref{['eq:PbLambda']} in $\lambda$, depicted in different shades of purple for various values of $\gamma$, according to Theorem \ref{['th:exist_gam_fix']}, Theorem \ref{['th:lambda_bh']} and Theorem \ref{['th:limit_gamma']}. The solution set of the associated Neumann problem is highlighted in red, while, as long as it exists, the solution set of the Dirichlet problem \ref{['eq:PbDirich']} is shown in blue.
• Figure 2: Sketch of the different sets that appear in the proof of Proposition \ref{['lem:conv_gam_fix']}.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof