Variational approach for the n-dimensional stationary Navier-Stokes equations with a damping term

We study the \(n\)-dimensional stationary Navier-Stokes equations with a damping term by developing a new general minimax principle. This principle is sufficiently broad to be applied in various contexts, and here it is used to establish the existence of weak solutions for both linear and nonlinear damping, without restrictions on the damping constant. The damping term, which models physical effects such as porous media flow, drag, friction, and dissipation, also provides a mathematical advantage by improving the regularity of solutions compared to the classical Navier-Stokes system. Our results cover the cases of positive and negative damping constants and yield existence theorems under different ranges of \(p\) and spatial dimensions. In particular, we prove solvability even in borderline situations, such as when \(\mu = -\lambda_1\), where coercivity is lost and traditional minimax arguments typically fail. The general minimax framework we introduce is flexible and can be adapted to other nonlinear PDEs, especially when symmetry or structural properties are involved. For more information see https://ejde.math.txstate.edu/Volumes/2026/01/abstr.html