On degenerate nonself-adjoint differential equations of fourth order

We consider the degenerate nonself-adjoint differential equation of fourth order $Lu\equiv(t^{\alpha}u^{\prime\prime})^{\prime\prime}+au^{\prime\prime\prime}-pu^{\prime}+qu=f$ where $t\in(0, b), \ 0\leq\alpha\leq 2, \alpha\neq 1, a, p, q $ are the constant numbers and $a\neq0, p>0, f\in L_2(0, b)$. We prove that the statement of the Dirichlet problem for the above equation depends on the sign of the number $a$ (Keldysh Teorem).