Uniqueness of the strong positive solution for a general quasilinear elliptic problem with variable exponents and homogeneous Neumann boundary conditions using a generalization of the $p(x)$-D\'{i}az-Saa inequality

Abstract

In this paper, we study a generalization of the D\'iaz-Saa inequality and its applications to nonlinear elliptic problems. We first present the necessary hypotheses and preliminary results before introducing an improved version of the inequality, which holds in a broader functional setting and allows applications to problems with homogeneous Neumann boundary conditions. The significance of cases where the inequality becomes an equality is also analyzed, leading to uniqueness results for certain classes of partial differential equations. Furthermore, we provide a detailed proof of a uniqueness theorem for strong positive solutions and illustrate our findings with two concrete applications: a multiple-phase problem and an elliptic quasilinear equation relevant to image processing. The paper concludes with possible directions for future research.