Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L¹(Ω). Furthermore the weight function θ(·) is in W̊^{1,p→(·) (Ω)}, with θ(·) > 0 and connected with the coefficient b(·) ∈ L¹(Ω) of the lower order term.