Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian

TL;DR

This paper investigates conditions for the existence of nonnegative bounded solutions to fractional semilinear elliptic equations in the entire space, focusing on the fractional Laplacian operator and specific nonlinearities.

Contribution

It provides necessary and sufficient conditions for solutions to fractional semilinear equations involving the fractional Laplacian in \\mathbb{R}^N.

Findings

Characterization of solution existence based on the properties of \\rho(x) and \\varphi(u)

Establishment of criteria linking the nonlinearity and the fractional Laplacian

Extension of classical results to fractional elliptic equations in unbounded domains.

Abstract

The main goal is to establish necessary and sufficient conditions under which the fractional semilinear elliptic equation $\Delta^{\frac{\alpha}{2}} u=\rho(x)\,\varphi(u)$ admits nonnegative nontrivial bounded solutions in the whole space $\mathbb{R}^N$.