Uniqueness of radial solutions for $m$-Laplacian equations in low dimensions

TL;DR

This paper proves the uniqueness of radial solutions for m-Laplacian equations in low dimensions, extending previous results to the case where the dimension is less than or equal to m, and resolves an open problem in the field.

Contribution

It provides a complete uniqueness theory for radial solutions of m-Laplacian equations in all dimensions, including the low-dimensional case N ≤ m, which was previously unaddressed.

Findings

Established uniqueness of solutions for N ≤ m under certain conditions.

Extended the range of nonlinearities for which uniqueness holds.

Resolved an open problem posed by Pucci and Serrin.

Abstract

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.