On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f ( x, s ) s −1 → a ∈)0, ∞) as s →+∞; and (ii) f ( x, s ) s –1 is non decreasing as a function of s ≥ 0, a.e. x → R N .