Liouville-type theorems for real sub-Laplacians

An inequality generalizing the classical Liouville and Harnack Theorems for real sub-Laplacians ℒ is proved. A representation formula for functions $u$ for which ℒu is a polynomial is also showed. As a consequence, some conditions are given ensuring that u is a polynomial whenever ℒu is a polynomial. Finally, an application of this last result is given: if ψ is a C ² map commuting with ℒ, then any of its component is a polynomial function. Received: 3 November 2000     

Liouville-type theorems for semilinear elliptic systems

In this paper, we provide two Liouville-type theorems for some semilinear elliptic systems. Our proofs are based on the moving spheres method and the general Kelvin transformed function. Four previously established results by different authors as consequence are obtained.     

Liouville-type theorems for subharmonic functions on complete manifolds

Applying the generalized maximum principle, a Liouvilletype theorem of subharmonic functions on complete Riemannian manifolds is shown and a Liouville-type differential inequality on properly immersed complete submanifolds is given. Project supported by the National Natural Science Foundation of China and K.C. Wong Educational Fund     

Liouville-type theorems for some complex hypoelliptic operators

It is shown that the natural analogue of Liouville's theorem holds for the well-known hypoelliptic operators _α (for ¦α¦ < n) introduced and studied by Folland and Stein on the Heisenberg group H_n. Since these operators are non-real for α ≠ 0, the usual methods of potential theory fail and are replaced by an explicit use of the fundamental solution.     

Liouville-type theorems for nonlinear degenerate parabolic equation

We study Liouville-type theorems for degenerate parabolic equation of the form \({u_t-{\rm div}(|\nabla u|^{m-2}\nabla u) = u^p}\) where \({m > 2}\) and \({p > m - 1}\). We prove the optimal Liouville-type results in dimension \({N = 1}\), and for radial solutions in any dimension. We also provide some partial results for non-radial solutions in dimension \({N \geq 2}\). Our proofs are based on a generalized Gidas–Spruck technique, combined with the idea of Serrin and Zou (Acta Math 189(1):79–142, 2002) and of Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198, 1998). Finally, we clarify and correct some of the previous results on this topic.     

Liouville-type theorems for a nonlinear fractional Choquard equation

In this paper, we are concerned with the fractional Choquard equation on the whole space ${\mathbb{R}}^N$ $${(-\mathrm{\Delta })}^{s}u=\left(\frac{1}{{|x|}^{N-2s}}\ast u^p\right)u^{p-1}$$ with , $N>2s$ and $p\in \mathbb{R}$. We first prove that the equation does not possess any positive solution for $p\le 1$. When $p>1$, we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.     

Some Liouville-type theorems for harmonic functions on Finsler manifolds

In this paper, we give some Liouville-type theorems for L^p$L^p$(p∈R)$(p\in \mathbb{R})$ harmonic (resp. subharmonic, superharmonic) functions on forward complete Finsler manifolds. Moreover, we derive a gradient estimate for harmonic functions on a closed Finsler manifold. As an application, one obtains that any harmonic function on a closed Finsler manifold with nonnegative weighted Ricci curvature _RicN${\mathit{Ric}}_N$(N∈(n,∞))$(N\in (n,\infty ))$ must be constant.     

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.     

Singularity and decay estimates for the conformal k-hessian equation via Liouville-type theorems

We study some connections between Liouville type theorems and local properties of nonnegative solutions to conformal k-hessian equations by making use of an elementary lemma for all positive functions in Li and Zhang (J. Anal. Math. 90 (2003), 27–87) and related Liouville type theorems in Li and Li (Acta. Math. 195 (2005), 117–154). Research of the second author is supported by Tianyuan Fund of Mathematics (10826061).     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity

We study the existence versus absence of nontrivial weak solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing lower order nonlinearities with potentials decaying to zero at infinity.