A study of some nonlinear elliptic differential equations in 𝐿 𝑚 (Ω)

Abstract

This thesis is devoted to the study of some classes of nonlinear and anisotropic elliptic differential equations with data belonging to Lebesgue spaces 𝐿 𝑚 (Ω). In the first part, we recall the essential preliminaries, including properties of classical Lebesgue and Sobolev spaces, convergence theorems, and fundamental topological tools such as the Faedo-Galerkin method and Minty’s lemma. The second part focuses on the existence and uniqueness of weak solutions for nonlinear elliptic equations in bounded domains, utilizing the properties of pseudo-monotone, hemicontinuous, and coercive operators. Finally, we investigate the framework of anisotropic elliptic equations, establishing the priori estimates and regularity of solutions under nonisotropic Sobolev embeddings when the data is locally integrable or lies in 𝐿 𝑚 (Ω).