On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on <i>ℝ^N</i>

TLDR

The authors aim to use a generic theorem to construct a special Palais–Smale sequence with additional properties that guarantee convergence, and to apply this to prove the existence of a positive solution for a functional with mountain‑pass geometry. They derive the generic theorem via Struwe’s monotonicity trick and then employ it to build a special Palais–Smale sequence for a given functional, ensuring its convergence. The theorem shows that for most functionals with mountain‑pass geometry a bounded Palais–Smale sequence exists at the MP level, and the authors apply this to nonlinearities satisfying specified growth and monotonicity conditions, obtaining a positive solution on ℝⁿ.

Abstract

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 .