The article is dedicated to the investigation of existence of weak solutions to a quasilinear elliptic equation involving terms with variable exponents –d/dxi  ai (x, u, ∇u) – λb (x) |u|p(x) − 2 u = f (x, u) in the variable exponential Sobolev space. Applying the Galerkin method for pseudo-monotone operators, we establish conditions under which there exists at least one weak solution to the boundary problem for such equations. Also, we prove the uniqueness of the weak solution under some additional monotony conditions on operator A0 and function f in functional Sobolev space.