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

Abstract

Picone identity is a powerful tool for proving qualitative properties of differential operators with ubiquitous applications in the analysis of partial differential equations, so generalizing it for different types of differential equations has become a desired venture. p(x)-Laplacian is a non-homogeneous quasilinear partial differential operator arising from various mathematical model with non-standard growth. However, in this paper, we establish a new generalized nonlinear variable exponent Picone identities for p(x)-sub-Laplacian. As applications we prove uniqueness, simplicity, monotonicity 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 Caccioppoli estimates with variable exponents.