Regularly varying solutions of subhomogeneous differential equations with p(t)-Laplacian

Abstract

This paper investigates the asymptotic behavior of increasing solutions to subhomogeneous differential equations involving the p(t)-Laplacian operator. Specifically, we consider the quasilinear equation (a(t)|y'|(p(t)) sgny')' = b(t)|y|(q(t)) L-G(|y|)sgny where p(t) and q(t) are variable exponents and L-G is a slowly varying perturbation. Our focus is on regularly varying solutions under the subhomogeneity condition p(t) > q(t) for large t. We show that all increasing solutions are regularly varying, derive asymptotic formulas for these solutions, and demonstrate their examples. This work contributes to the understanding of nonoscillatory solutions and shows how regular variation can be useful in studying differential equations involving variable exponents.