Correct and coercive solvability conditions for a degenerate high order differential equation

In the work, we consider a fifth-order singular differential equation with variable coefficients. The singularity means, firstly, that the equation is given on the real axis $\mathbb{R}=(-\infty,\infty)$, and secondly, its coefficients are unbounded functions. We study a new degenerate case, when the intermediate coefficients of the equation grow faster than the lowest coefficient (potential), and also the potential is not sign-definite. We obtain sufficient conditions for the existence and uniqueness of the generalized solution of the equation. We also prove a coercive estimate for the solution. The coefficients of the equation are assumed to be smooth, but we do not impose any restrictions on their derivatives to prove the results. Note that the well-known stationary Kawahara equation can be reduced to the considered equation after linearization.