Landesman-Lazer condition revisited: the influence of vanishing and oscillating nonlinearities

TL;DR

This paper revisits the Landesman-Lazer condition for semilinear problems at resonance, proposing a generalized criterion that includes cases with vanishing and oscillating nonlinearities, broadening the applicability of existence results.

Contribution

It introduces a new sufficient condition for weak solutions that extends the classical Landesman-Lazer condition to more complex nonlinear behaviors.

Findings

Established a generalized Landesman-Lazer condition

Proved existence of weak solutions under new criteria

Included cases with vanishing and oscillating nonlinearities

Abstract

In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of the asymptotic properties of nonlinearity. Our condition generalizes the classical Landesman-Lazer condition but it also covers the cases of vanishing and oscillating nonlinearities.