The solvability results for the third-order singular non-linear differential equation

Abstract

We give some conditions for solvability in L₂(ℝ) (ℝ = (-∞, +∞)) of the following singular non-linear differential equation:

ly ≡ -y″″″(x) + q(x, y, y′) y′ + s(x, y, y′) y = h(x).

We assume that q and s are real-valued unbounded functions and q does not obey the “potential” s. For the solution y we prove that

‘‘y″″″’’₂ + ‘‘q(∙, y, y′) y′’’₂ + ‘‘s(∙, y, y′) y’’₂ < ∞,

where ‘‘·’’₂ is the norm in L₂. To establish these facts, we use coercive solvability results for the corresponding linear third-order differential equation obtained by us earlier.