A fractional Landesman-Lazer type problem set on R^N

TL;DR

This paper proves the existence of positive solutions for a fractional Landesman-Lazer problem on R^N using an abstract monotonicity-trick, even when the nonlinearity does not meet the usual growth conditions.

Contribution

It introduces an approach leveraging Struwe's monotonicity-trick to handle fractional problems without the Ambrosetti-Rabinowitz condition.

Findings

Existence of positive solutions established

Applicable to fractional Laplacian problems on R^N

Handles nonlinearities without standard growth conditions

Abstract

By using the abstract version of Struwe's monotonicity-trick we prove the existence of a positive solution to the problem (-\Delta)^s u + K u = f(x, u) in R^N u\in H^s (R^N), K>0 where f(x, t): R^N\times R \rightarrow R is a Caratheodory function, 1-periodic in x and does not satisfy the Ambrosetti-Rabinowitz condition.