This paper extends the uniqueness results of Serrin and Tang [\textit{ Indiana Univ. Math. J.}, 49 (2000), pp. 897--923] to the low-dimensional case $1\leq N\leq m$ with $m>1$. We consider radial solutions of the overdetermined problem \[ \begin{ cases} -\Delta_m u = f(u), \quad u>0 & \text{ in } B_R,\\[4pt] u = \partial_\nu u = 0 & \text{ on } \partial B_R, \text{ if } R<\infty,\\[4pt] \displaystyle\lim_{ |x|\to\infty} u(x)=0, & \text{ if } R=\infty, \end{ cases} \] 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 analysis is motivated by connections to sharp forms of the Gagliardo--Nirenberg and Nash inequalities. Although the overall framework follows that of Serrin and Tang, the details of our proofs differ substantially in the low-dimensional setting. In particular, Serrin and Tang explicitly noted that their techniques rely heavily on the condition $N>m$ and do not readily extend to $N\leq m$ (see Subsection~6.2 of their work). The present paper closes this gap, thereby providing a complete uniqueness theory for all dimensions. As a concrete example, for the canonical nonlinearity $f(u) = -u^p + u^q$ with $p<q$, our result covers the full range $-1 < p < q < m^*-1$, where $m^*:=\frac{ Nm}{ N-m}$ for $N>m$ and $m^* = \infty$ for $N\leq m$. Consequently, our work also completely resolves an open problem posed by Pucci and Serrin [\textit{ Indiana Univ. Math. J.}, 47 (1998), pp. 501--528], which had been settled for $N>m$ in the earlier work of Serrin and Tang.
Categories: math.AP