$p(x)$-stability of the Dirichlet problem for Poisson's equation with variable exponents

We investigate the structural stability of solutions to boundary value problems for the variable exponent $p(x)$-Laplacian. Stability questions for such problems under perturbations of the boundary operator, the differential operator, boundary data, or the domain have a long history and play a central role in the analysis of nonlinear partial differential equations (PDEs). In this work, we consider the Poisson boundary value problem with nonhomogeneous boundary conditions and study the behavior of its solutions under variations of the exponent functions $p(x)$. Our results extend the classical stability theorem of Lindqvist (1987), originally formulated for constant $p$, to the variable exponent setting. Moreover, our approach sharpens and generalizes the framework developed by Zhikov (2011), allowing for nonhomogeneous boundary data and providing stronger convergence results for the associated family of solutions. Specifically, it is shown that if the sequence $(p_j(x))$ increases uniformly to $p(x)$ in a bounded, smooth domain $\mathrm{\Omega }$, then the sequence $(u_i)$ of solutions to the Dirichlet problem for the $p_i(x)$-Laplacian with fixed boundary datum $\phi$ converges (in a sense to be made precise) to the solution $u_p$ of the Dirichlet problem for the $p(x)$-Laplacian with boundary datum $\phi$. A similar result is proved for a decreasing sequence $p_j\searrow p$.