A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general H\'enon-type superquadratic and gradient growth

TL;DR

This paper develops a unified framework to classify Liouville properties and construct nontrivial solutions for fractional elliptic equations with Hénon-type growth, revealing critical balances and solution behaviors across different regimes.

Contribution

It introduces a comprehensive analytical approach to identify critical parameters and construct explicit solutions for nonlocal elliptic equations with general growth conditions.

Findings

Critical balance    separates solution regimes.

Supercritical case: solutions are constant.

Subcritical case: explicit positive radially symmetric solutions.

Abstract

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^{\gamma}\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric, translation-invariant, uniformly elliptic integro--differential operator of order $2s\in(0,2)$, and $H,G$ satisfy general structural and growth conditions. A unified analytical framework is developed to identify the precise critical balance $\gamma+p=2s$, which separates the supercritical, critical, and subcritical situations. In the supercritical case $\gamma+p>2s$, the diffusion dominates the nonlinear term and every globally defined solution with subcritical growth must be constant; in the critical case $\gamma+p=2s$, all bounded positive solutions are constant, showing that the nonlocal diffusion prevents the formation of nontrivial equilibria; in the subcritical case $\gamma+p<2s$, we are able construct a unique, positive, radially symmetric, and monotone entire solution with explicit algebraic decay \[ u(x)\sim (1+|x|^2)^{-\beta}, \qquad \beta=\frac{2s+\gamma-p}{1-p}>0. \] The proofs rely on new nonlocal analytical techniques, including quantitative cutoff estimates for general integro--differential kernels, a fractional Bernstein-type transform providing pointwise gradient control, and moving plane and sliding methods formulated in integral form to establish symmetry and uniqueness. The current investigation provides an equivalent and unifying contribution to Liouville properties and related existence results, comparable to the recent deep studies of Chen--Dai--Qin~\cite{Chen2023} and Biswas--Quaas--Topp~\cite{Biswas2025} on this direction.