Boundary Value Problem for Fourth Order Inhomogeneous Equation with Variable Coefficients

We study the boundary value problem for a fourth order inhomogeneous equation with lower order terms and variable coefficients in a rectangular domain. We find sufficient conditions on the coefficients of the equation guaranteeing the uniqueness of a solution. We give an example of coefficients satisfying these conditions and show that the homogeneous problem can have a nontrivial solution if these conditions fail. We express the solution in terms of the Green function.