Existence and summability of solutions to nonlinear X-elliptic equations with measurable coefficients

We prove an existence result for solutions to a class of nonlinear degenerate elliptic equations with measurable coefficients and zero Dirichlet boundary condition. The main term is given by a nonlinear operator in divergence form associated to a family of vector fields which satisfy a Poincaré inequality and the doubling condition. Furthermore, we prove that the solutions satisfy a generalization of the $$L^p$$ -regularity results which hold for the solutions to Leray–Lions type equations.