Nonlinear variable exponent Picone identity and $p(x)$-sub-Laplacian first eigenvalue for general vector fields

TL;DR

This paper develops a generalized nonlinear Picone identity for the $p(x)$-sub-Laplacian, leading to new insights into the eigenvalues and inequalities associated with variable exponent operators on general vector fields.

Contribution

It introduces a novel nonlinear Picone identity for $p(x)$-sub-Laplacian and applies it to establish properties of the first eigenvalue and related inequalities.

Findings

Proved uniqueness and simplicity of the first eigenvalue.

Established monotonicity and isolatedness of the first eigenvalue.

Derived Hardy type inequalities and Caccioppoli estimates.

Abstract

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.