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

TL;DR

The paper addresses the existence of positive solutions to the nonlocal fractional Schrödinger-type equation $(-\Delta)^s u + Ku = f(x,u)$ in $\mathbb{R}^N$ with $s\in(0,1)$, $N>2s$, and a nonlinearity $f$ that is $1$-periodic in $x$ and does not satisfy the Ambrosetti-Rabinowitz condition. It adopts a variational framework and applies the abstract Struwe monotonicity trick (via a family of functionals $I_\lambda$) to obtain bounded Palais–Smale sequences for a.e. $\lambda \in [1,2]$, circumventing AR and nonlocal challenges. Using translation invariance and weak maximum principles, it constructs a nontrivial, nonnegative limit, which yields a positive solution for $\lambda=1$, with further regularity and decay at infinity. The results extend variational methods to nonlocal operators with periodic nonlinearities under relaxed growth conditions, highlighting the applicability to fractional Schrödinger equations without AR.

Abstract

By using the abstract version of Struwe's monotonicity-trick we prove the existence of a positive solution to the problem (-Δ)^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.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Lemma 1
• Lemma 2
• Lemma 3
• Definition 1
• Lemma 4