Differentiability of the Energy Functional of a Class of Quasi-linear Elliptic

To study the differentiability of a class of quasi-linear elliptic equations energy functional by using the embedding theorems and some other properties of generalized Sobolev spaces. According to the variation principle the energy functional of equations is expressed in some appropriate generalized Sobolev spaces, the various parts of differentiability of the energy functional of equations are discussed under certain conditions. Using Lebesgue dominated convergence theorem and embedding theorem, the energy functional of equations is proved. Finally, the overall differentiability is proved. The conclusions lay the foundation for the next step to prove the existence of the critical point of the energy functional, that is, the existence of solutions of the equations.