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

TL;DR

This work provides a unified Liouville theory for nonlinear nonlocal elliptic equations with weighted and gradient-type nonlinearities, identifying the critical balance $\gamma+p=2s$ that separates regimes of rigidity and existence. The authors develop a fractional Bernstein transform and nonlocal maximum principles to obtain gradient controls and rigidity in the supercritical and critical cases, and construct explicit radially symmetric subcritical solutions with precise decay in the subcritical regime. They further establish radial symmetry and uniqueness for bounded positive solutions via integral moving planes, and develop barrier methods to obtain existence in the subcritical range, including sharp two-sided decay profiles $u(x)\sim (1+|x|^2)^{-\beta}$ with $\beta=\frac{2s+\gamma-p}{1-p}>0$. The results unify and extend recent Liouville-type analyses for nonlocal problems, providing a framework applicable across polynomial, logarithmic, exponential, and singular nonlinearities, and connect to broader themes in fractional diffusion and nonlocal geometry.

Abstract

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^γ\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] 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 $γ+p=2s$, which separates the supercritical, critical, and subcritical situations. In the supercritical case $γ+p>2s$, the diffusion dominates the nonlinear term and every globally defined solution with subcritical growth must be constant; in the critical case $γ+p=2s$, all bounded positive solutions are constant, showing that the nonlocal diffusion prevents the formation of nontrivial equilibria; in the subcritical case $γ+p<2s$, we are able construct a unique, positive, radially symmetric, and monotone entire solution with explicit algebraic decay \[ u(x)\sim (1+|x|^2)^{-β}, \qquad β=\frac{2s+γ-p}{1-p}>0. \] 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.

Theorems & Definitions (35)

• Theorem 1.1: Unified Liouville theorem for gradient-type nonlinearities for supercritical and critical cases
• Theorem 1.2: Existence, radial symmetry, and monotonicity in the subcritical case
• Lemma 2.1: Nonlocal maximum principle
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Cut-off estimate
• proof
• Proposition 2.4: Bernstein transform and gradient estimate
• proof