An existence result for anisotropic quasilinear problems

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p$-Laplacian operator with a coefficient that is locally bounded inside the domain and satisfying certain additional integrability assumptions. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value problem with a non-linearity $f\left(u\right)$ satisfying $f\left(0\right)\le 0$ and having $(p-1)$-sublinear growth at infinity.