  We establish the existence of positive solutions for a nonlinear elliptic
Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The
nonlinearity considered depends on the gradient of the unknown function and an
exponential term. In such case, variational methods cannot be applied. Our
approach is based on approximation scheme, where we consider a new class of
normed spaces of finite dimension. As a particular case, we extended the result
achieved by De Araujo and Montenegro [2016] for any $N>2$.
