A uniqueness result for two-step indefinite $p$-Laplacian equations

Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \begin{equation*} u’‘+a(t)u^γ=0 \end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $\gamma \in \mathbb{R}\setminus \{-1,0,1\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $\gamma =-1$. More generally, for every $p>1$, we consider the equation \begin{equation*} \left(\left\lvert u’\right\rvert^{p-2}u’\right)'+a(t)u^γ=0 \end{equation*} under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all $\gamma \in \mathbb{R}\setminus \{0,p-1\}$. The necessary mean-value condition $\gamma {\int }_0^{T}a(t)\mathrm{d}t<0$ is also sufficient for existence when $\gamma \le -1$ or $\gamma >p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $\gamma \in (-1,p-1)\setminus \{0\}$. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.