Examining the solution of Steklov boundary value problem involving the p(x)-Laplace operator

In this thesis, the existence of a nontrivial weak solution to the Steklov boundary value problem involving the 𝑝(𝑥)−Laplacian operator is investigated in the variable exponent Sobolev space. The variational approach method is employed in the exploration of these solutions. In the first section, information is given about the historical development of variable exponent spaces that we initially worked on. In addition, the existence of solutions to problems with different boundary value conditions involving the 𝑝(𝑥)−Laplacian operator under the variational approach, their multiplicity, the existence of at least one weak solution, the existence of at least one non-zero weak solution, the existence of infinite solutions or the existence of at least three solutions are examined. Some of the studies are given. Finally, the problem we will work on and the method to be used are defined. In the second section, information about some basic definitions and theorems in the thesis is given. Then, information is given about the definition of variable exponent Lebesgue space and variable exponent Sobolev space, which are the spaces where we search for solutions to our problem, and the theorems used in these spaces. In the third section, information is given about the variational approach, which is the method used when investigating the existence of solutions to problems with different boundary value conditions, including standard and non-standard growth conditions. Some theorems needed when using this approach are included. In addition, some definitions necessary for these theorems are defined. Then, information is given about some studies using these theorems. In the fourth section, the original contribution of the thesis is presented. This section focuses on the weighted elliptic Steklov problem with nonstandard growth conditions. Under suitable conditions, the existence of at least one nontrivial weak solution to the problem is explored in the variable exponent Sobolev space using both the Mountain-Pass theorem and the Ambrosetti-Rabinowitz condition. Keywords: 𝑝(𝑥)− Laplacian, Steklov problem, Variational approach, Mountain-Pass theorem, Ambrosetti-Rabinowitz condition, Weak solution, Steklov problem.