On some uniqueness results
Patrizia Pucci, Jianjun Zhang, Xuexiu Zhong
TL;DR
This work extends the uniqueness results of Serrin–Tang for radial solutions of the overdetermined problem $- abla_m u=f(u)$ in a ball to the low-dimensional regime $1\le N\le m$ with $m>1$, under assumptions that $f$ changes sign at $b$ and $g(u)=\frac{u f'(u)}{f(u)}$ is nonincreasing for large $u$. The authors deploy an inverse-function method, transforming the radial ODE into an equation for $r=r(u)$ and introducing the auxiliary quantities $A(u)$ and $P(u)$ to obtain monotonicity control and a rigorous comparison framework. In the one-dimensional case ($N=1$) they prove uniqueness via a direct inversion argument (or an integral identity yielding $F(\alpha)=0$), while for $2\le N\le m$ they develop a sophisticated sign-analysis of auxiliary functions $t,s,T,S$ and a contradiction argument based on carefully constructed functionals $P_i$; under the stated hypotheses the radial Dirichlet–Neumann free boundary problem admits at most one radial solution. This provides the needed foundation to justify sharp Gagliardo–Nern Nash-type inequalities in all dimensions and clarifies the role of the dimension range $N\le m$ in uniqueness theory for $m$-Laplacian problems.
Abstract
This paper aims to extend the results of Serrin and Tang in [{\it Indiana Univ. Math. J., 49 (2000), 897--923}] to the low-dimensional case. Specifically, the paper deals with the radial solutions of the following overdetermined problem $$ \begin{cases} -Δ_m u=f(u),\quad u>0~\hbox{in}~B_R,\\ u=\partial_νu=0~\hbox{on}~\partial B_R, \end{cases} $$ where $B_R$ is the open ball of $\mathbb{R}^N$ centered at 0 and with radius $R>0$. We prove uniqueness when $1\leq N\leq m$ {and $m>1$} under certain suitable assumptions on~$f$. Additionally, this work is motivated by the sharp Gagliardo-Nirenberg/Nash inequality. While the framework presented in this article is standard and closely resembles that of Serrin and Tang, the detail of our proofs differ significantly. It is important to note that Serrin and Tang explicitly stated (see Subsection~6.2 of their work) that {\it``the proofs in the present paper rely extensively on the assumption $N>m$ and cannot be extended easily to values $N\leq m$."}