Study of fractional semipositone problems on $\mathbb{R}^N$

TL;DR

This paper investigates nonlocal fractional semipositone problems on al Rb2N, proving existence of solutions near zero parameter using variational methods and establishing uniform estimates for positivity.

Contribution

It introduces new existence results for fractional semipositone problems with subcritical growth and weaker conditions, using a novel Brezis-Kato type estimate.

Findings

Existence of mountain pass solutions near zero parameter.

Uniform rezis-Kato0 estimates ensuring positivity.

Applicability to problems with subcritical growth and weaker nonlinearities.

Abstract

Let $s \in (0,1)$ and $N >2s$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-\Delta)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.