Investigation of solutions of the Steklov boundary value problem involving nonstandard growth condition

In this thesis, the existence of a weak nontrivial solution solution of a Steklov boundary value problem with non-standard growth condition is investigated in variable exponent Sobolev space using a variational approach. In the first chapter, information is given about the historical development of the variable exponent Lebesgue space and the variable exponent Sobolev spaces, which are the spaces in which the solution of our problem is investigated. Then, in recent years, some studies with different boundary conditions in these spaces have been examined by using the variational approach. After that, the Steklov boundary value problem, which includes the non-standard growth condition that we will work on, is given. In the second chapter, information about the basic definition and theorem that will be used in other chapters is given. Then, information about the basic definitions, theorems and notations about the variable exponent Lebesgue spaces we have studied is given. In addition, after defining the variable exponent Sobolev space, information about the necessary definitions, theorems and notations about this space is given. In the third chapter, after giving information about the variational approach, some theorems used in the variational approach are given to show the existence of solutions to boundary value problems involving standard and non-standard growth conditions. In this part of our study, some definitions necessary to use theorems are given. In the last part, information about some studies carried out under the variational approach is given by using some special definitions together with these theorems. The fourth chapter constitutes the original part of the thesis. In this section, the existence of a non-zero weak solution of the Steklov boundary value problem involving non-standard growth condition is shown by using the Mountain-Pass theorem together with the Ambrosetti-Rabinowitz condition under the variational approach. While investigating this solution, the energy functional corresponding to the problem was used as a tool. The weak solution is investigated in variable exponent space.