  In this paper, we establish a new generalized nonlinear variable exponent
Picone identities for $p(x)$-sub-Laplacian. As applications we prove
uniqueness, simplicity, momotonicity and isolatedness of the first nontrivial
Dirichlet eigenvalue of $p(x)$-sub-Laplacian with respect to the general vector
fields. Further applications yield Hardy type inequalities and Caccioppolli
estimates with variable exponents.

Analysis of PDEs