Revistes Catalanes amb Accés Obert (RACO)

Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations

G. Lu

Resum


This paper proves Harnack's inequality for solutions to a class of quasilinear subelliptic differential equations. The proof relies on various embedding theorems into nonisotropic Lipschitz and BMO spaces associated with the vector fields $X_{1},\ldots, X_{m}$ satisfying Hörmander's condition. The nonlinear subelliptic equations under study include the important p-sub-Laplacian equation, e.g.,
$$
\sum_{j=1}^{m}X_{j}^{*}\left(|Xu|^{p-2}X_{j}u\right) =A|Xu|^{p}+B|Xu|^{p-1}+C|u|^{p-1}+D,\\ 1 < p < \infty
$$
where $|Xu|=\sum_{j=1}^{m}\left(|X_{j}u|^{2}\right)^{\frac{1}{2}}$ and $A$ is a constant; $B$, $C$ and $D$ can be in appropriate function spaces. We note that $A$ can be nonzero.

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