题目：Uniqueness of radial solutions for m-Laplacian equations in low dimensions
摘要：We extend the uniqueness results of Serrin and Tang[IUMJ,2000] to the low dimensional case $1\leq N\leq m$ with $m>1$. We consider radial solutions of the overdetermined problem
$$
\begin{cases}
-\Delta_m u = f(u), \quad u>0 & \text{in } B_R,\\
u = \partial_\nu u = 0 & \text{on } \partial B_R, \text{ if } R<\infty,\\
\lim_{|x|\to\infty} u(x)=0, & \text{if } R=\infty,
\end{cases}
$$
where $B_R$ is the open ball in $\mathbb{R}^N$ centered at the origin with radius $R>0$ (the case $R=\infty$ corresponds to the whole space, for studying positive ground states). Under suitable assumptions on the nonlinearity $f$, we establish the uniqueness of such solutions, whenever they exist.
Our results completely resolve the open problems posed by Serrin–Tang and by Pucci–Serrin. This is joint work with Professors Patrizia Pucci and Jianjun Zhang, and it is to appear in IUMJ (2026).
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