Existence of a Local Renormalized Solution of an Elliptic Equation with Variable Exponents in $\mathbb{R}^n$

The article is devoted to the study of second-order quasilinear elliptic equations with variable nonlinearity exponents and a locally integrable right-hand side in the space $\mathbb{R}^n$. The author adapts the concept of a locally renormalized solution for equations with variable growth exponents, generalizing the results of M. F. Bidaut-V\'{e}ron and L. V\'{e}ron obtained for equations with constant exponents. The work establishes conditions on the structure of the quasilinear elliptic operator with variable growth that are sufficient for the correct definition of a locally renormalized solution. The author derives a priori local estimates characterizing the regularity of the solution and, based on these, proves the existence of a locally renormalized solution in the space $\mathbb{R}^n$ without additional restrictions on its growth at infinity. Furthermore, the work demonstrates that for a non-negative right-hand side, the solution is also non-negative almost everywhere. The research employs methods of functional analysis, including the theory of Lebesgue and Sobolev spaces with variable exponents. The proofs are based on compactness and monotonicity techniques, as well as the use of special test functions. The results of the work are significant for the theory of nonlinear elliptic equations and can be applied to further studies of degenerate equations and problems with measure-valued data. The study contributes to the development of analytical methods for equations with variable nonlinearity exponents and expands the applicability of the concept of locally renormalized solutions.