Finsler N-Laplace equations with critical exponential nonlinearity

In this article, we investigate some anisotropic equations involving the Finsler N-Laplace operator ${\Delta }_{H,N}$, with Trudinger-Moser type critical exponential nonlinearity which involves the anisotropic norm. More precisely, we establish the existence of nontrivial solutions to the problem$-{\Delta }_{H,N}u=f(x,u),u\ge 0\text{ in }\Omega ,$ on a smooth bounded domain $\Omega \subset R^N,N\ge 2$, and to the anisotropic quasilinear Schrödinger problem$-{\Delta }_{H,N}u+V\left(x\right)|u{|}^{N-2}u=g(x,u),u\ge 0\text{ in }{R}^N.$ Here the nonlinearities f and g exhibit critical exponential-type growth, and the potential V is a continuous function that retains the compactness of the associated Sobolev embedding into a larger class of Lebesgue spaces under suitable assumptions. We obtain the existence results by employing the anisotropic Trudinger-Moser inequality, mountain-pass theorem, and Ekeland's variational principle.