Three-spheres theorems for subelliptic quasilinear equations in Carnot groups of Heisenberg-type (Q2817007)

In this paper, the authors prove a version of the Hadamard three-circles theorem for subsolutions of the subelliptic quasilinear \(p\)-harmonic equation on H-type groups.NEWLINENEWLINELet \(G\) be an H-type Lie group [\textit{A. Kaplan}, Trans. Am. Math. Soc. 258, 147--153 (1980; Zbl 0393.35015)] with homogeneous dimension \(Q\) and with inner product \(\langle \cdot, \cdot \rangle\) on the Lie algebra \(\mathfrak{g}\). Let \(\nabla_0\) be the standard sub-gradient on \(G\). Suppose \(u\) is a subsolution of the \(p\)-harmonic equation, \(1 < p < \infty\); that is, \(u\) is continuous and satisfies \(\int \langle |\nabla_0 u|^{p-2} \nabla_0 u, \nabla_0 \phi \rangle\,dx \leq 0\) for all \(\phi \in C^\infty_0\).NEWLINENEWLINELet \(|\cdot|\) denote the Folland-Kaplan gauge norm \(|(x,z)| = (|x|^4 + 16|z|^2)^{1/4}\), which is homogeneous with respect to the dilation of \(G\). Let \(M(r) = \sup\{u(X) : |X| = r\}\) be the maximum value of \(u\) on a gauge-norm sphere of radius \(r\) centered at the identity. The main result of the paper is the following three-spheres theorem (stated in a simplified form): if \(p \neq Q\), then for any \(r_1 < r < r_2\), we have NEWLINE\[NEWLINEM(r) \leq M(r_1) \frac{r^c - r_2^c}{r_1^c - r_2^c} + M(r_2) \frac{r_1^c - r^c}{r_1^c - r_2^c}NEWLINE\]NEWLINE where \(c = \frac{p-Q}{p-1}\). Equality holds when \(u\) is radial.NEWLINENEWLINEIn the case \(p=Q\), the result reads NEWLINE\[NEWLINEM(r) \leq M(r_1) \frac{\log \frac{r_2}{r}}{\log\frac{r_2}{r_1}} + M(r_2) \frac{\log\frac{r}{r_1}}{\log\frac{r_2}{r_1}}.NEWLINE\]NEWLINENEWLINENEWLINEAs corollaries, the authors derive growth estimates for subsolutions and a Liouville-type theorem. The key tool is a strong maximum principle.