Quasilinear stationary problems with a quadratic gradient term having singularities

We study the homogeneous Dirichlet problem for some elliptic equations with a first-order term b(u, Du) which is quadratic in the gradient variable and singular in the u variable at a positive point. Moreover, the gradient term that we consider changes its sign at the singularity. While dealing with an appropriate concept of solution that gives sense to the equation at the singularity, we prove the existence of solutions for every datum belonging to a suitable Lebesgue space. Furthermore, we show that the solutions pass through the singularity when the data are large enough.