Solutions to the p(x)- Laplacian equations in variable Lebesgue Sobolev spaces by using nehari manifold approach and mountain pass theorem

In this thesis, using variational approach and applying Nehari Manifold method, Mountain Pass theorem and Symmetric Mountain Pass theorem the existence of the solutions of the p(x)- Laplacian type equations were investigated in the variable exponent Lebesgue and Sobolev spaces.In the present thesis, solutions of two different equations which have non-standard growth condition were obtained in variable exponent Lebesque and Sobolev spaces.In the first part, some information was given about the historical progress of variable exponent Lebesgue --Sobolev spaces and about the applications of differential equations with non-standard growth condition.Then, some account was also added about the studies carried out so far and the results obtained with regard to differential eqations with non-standard growth condition.In the second part, some other information was given about the Lebesque Sobolev spaces as well as the basic kowledge that we would use in the following parts.In the third part, some account was given about the basic theorems, basic definitions and variational approach used for the existence of solutions of equations with standard and non- standard growth conditions. In addition, some major studies, in which these theorems were used, were referred to.In the fourth part, for a semilinear elliptic problem that possesses the Dirichlet boundary value conditions with non- Standard growth condition, at least the existence of two positive solutions was obtained in variable exponent Sobolev space by using the Nehari Manifold Method through a variational approach.In the fifth part, the existence and multiplicity of the the solutions were obtained in variable exponent Sobolev space for the nonuniform elliptic Laplacian equation by using the Mountain Pass theorem and Symetric Mountain Pass theorem.