发表机构
Università degli Studi di Udine（乌迪内大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了两步不定权 p-Laplacian 方程在 Neumann 和周期边界条件下正解的唯一性猜想，并给出所有参数范围内的完整存在性与唯一性分类，数值例子显示非两步权可产生多解。
AI 中文摘要
Boscaggin、Feltrin 和 Zanolin 猜想：与方程 $u''+a(t)u^\gamma=0$（其中 $a(t)$ 为两步不定权函数）相关的 Neumann 问题和周期问题，对每个 $\gamma\in\mathbb{R}\setminus\{-1,0,1\}$ 至多有一个正解。我们证明了这个猜想，并通过纳入对数势情形 $\gamma=-1$ 完善了唯一性图景。更一般地，对每个 $p>1$，我们考虑在 Neumann 或周期边界条件下的方程 $(|u'|^{p-2}u')'+a(t)u^\gamma=0$，并对所有 $\gamma\in\mathbb{R}\setminus\{0,p-1\}$ 建立了完整的解的存在性与唯一性分类。当 $\gamma\leq-1$ 或 $\gamma>p-1$ 时，必要均值条件 $\gamma\int_{0}^{T}a(t)\\,\mathrm{d}t<0$ 也是存在性的充分条件；而在中间范围 $\gamma\in(-1,p-1)\setminus\{0\}$ 内，则需要关于权均值的额外尖锐条件。最后，一个数值例子表明，对于权函数具有相同单次变号但内部轮廓非恒定的 Neumann 问题，存在多个正解，这凸显了两步假设的作用。
英文摘要
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 $γ\in\mathbb{R}\setminus\left\{-1,0,1\right\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $γ=-1$. 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 $γ\in\mathbb{R}\setminus\left\{0,p-1\right\}$. The necessary mean-value condition $γ\int_{0}^{T}a(t)\,\mathrm{d} t<0$ is also sufficient for existence when $γ\leq-1$ or $γ>p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $γ\in(-1,p-1)\setminus\{0\}$. 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.
Comments22 pages, 2 figures
