Liouville Type Theorem Research Articles

GRADIENT ESTIMATES FOR POSITIVE EIGENFUNCTIONS OF THE $\mathcal {L}$ -OPERATOR ON CONFORMAL SOLITONS AND THEIR APPLICATIONS

Abstract We prove a local gradient estimate for positive eigenfunctions of the $\mathcal {L}$ -operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for $\mathcal {L} u=0$ , which improves the one of Li and Sun [‘Gradient estimate for the positive solutions of $\mathcal Lu = 0$ and $\mathcal Lu ={\partial u}/{\partial t}$ on conformal solitons’, Acta Math. Sin. (Engl. Ser.)37(11) (2021), 1768–1782]. We also consider applications where manifolds are special conformal solitons and obtain a better Liouville type theorem in the case of self-shrinkers.

Gradient estimates for Δpu + Δqu + a|u|s−1u + b|u|l−1u = 0 on a complete Riemannian manifold and applications

In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of non-negative solutions to the nonlinear elliptic equation [Formula: see text] defined on a complete Riemannian manifold [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are constants and [Formula: see text], with [Formula: see text], is the usual [Formula: see text]-Laplace operator. Under some assumptions on [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], we derive gradient estimates and Liouville-type theorems for non-negative solutions to the above equation. In particular, we show that, if [Formula: see text] is a non-negative entire solution to [Formula: see text] ([Formula: see text]) on a complete non-compact Riemannian manifold [Formula: see text] with non-negative Ricci curvature and [Formula: see text], and [Formula: see text] where [Formula: see text] then [Formula: see text] is a trivial constant solution.

Liouville-type theorem for higher order Hardy-Hénon type systems on the sphere

In this paper, we study Liouville type theorems for the positive solutions to the following higher order Hardy-Hénon type system involving the conformal GJMS operator on the sphere Sn. In order to study this we first employ the Mobius transform to transform the above Hardy-Hénon type system on the sphere Sn into a higher order elliptic system on Rn. Then, we show that every positive solution of the higher order elliptic system on Rn is a solution to the associated integral system on Rn by using polyharmonic average and iteration arguments. We use the method of moving planes in integral form to prove that there are no positive solutions for the integral system on Rn. Finally, together with the symmetry of the sphere Sn, we obtain the Liouville type theorem of the higher order Hardy-Hénon type system involving the GJMS operator on the sphere. The results of this paper are also new even for the Lane-Emden system on the sphere.

Liouville-type theorems for partial trace equations with nonlinear gradient terms

In this paper, we will study various Liouville-type theorems for partial trace equations with nonlinear gradient terms. Specifically, we will provide sufficient conditions for non-negative viscosity subsolutions of these equations in Rn to vanish identically. For a prototype of such equations, we will give necessary and sufficient conditions for non-negative subsolutions in Rn to be identically zero.

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Liouville type theorems for subelliptic systems on the Heisenberg group with general nonlinearity

In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group \(\mathbb{H}^{n}\) involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane–Emden system on the Heisenberg group \(\mathbb{H}^{n}\), which also appears to be a new result in the literature.

Liouville-type laws for quasilinear equations with absorption or source

In this paper we are concerned with the Liouville-type theorems and the radial supersolutions for the m-Laplacian equations with absorption or source ±|∇u|q=Δmu+fp(u). Here, q>0 and fp satisfies either fp(s)>γsp near zero or equals to γsp for a positive constant pair {p,γ}.

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

Liouville type theorems for a quasilinear elliptic differential inequality with weighted nonlocal source and gradient absorption terms

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

AbstractIn this article we study local and global properties of positive solutions of − Δmu = ∣u∣p−1u+M∣∇u∣q in a domain Ω of ℝN, with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.

Analysis of harmonic functions under lower bounds of N-weighted Ricci curvature with ε-range

The behavior of harmonic functions on Riemannian manifolds under lower bounds of the Ricci curvature has been studied from both analytic and geometric viewpoints. For example, some Liouville type theorems are obtained under lower bounds of the Ricci curvature. Recently, those results are generalized under lower bounds of the N-weighted Ricci curvature RicψN with N∈[n,∞]. In this paper, we present a Liouville type theorem for harmonic functions of sublinear growth and a gradient estimate of harmonic functions on weighted Riemannian manifolds under weaker lower bounds of RicψN with N<0. We also prove an Lp-Liouville theorem under lower bounds of RicψN with N∈[n,∞] in a different way from previous studies. Our results are obtained as a consequence of an argument under lower bounds of RicψN with ε-range, which is a unification of constant and variable curvature bounds. Among various methods considered for the analysis of harmonic functions, this paper focuses on methods using the Moser's iteration procedure.

Viscosity solutions to the inhomogeneous reaction–diffusion equation involving the infinity Laplacian

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian: [Formula: see text] where the continuous function [Formula: see text] satisfies [Formula: see text] a positive function [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text]. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For [Formula: see text] and the non-positive inhomogeneous term [Formula: see text] we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on [Formula: see text] and [Formula: see text] we obtain the optimal [Formula: see text] regularity across the free boundary [Formula: see text] Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case [Formula: see text]

Liouville-type results for semilinear inequalities involving the Dunkl Laplacian operator

Let Δ k be the Dunkl generalized Laplacian operator associated to a root system R of R N and a nonnegative multiplicity function k defined on R and invariant by the finite reflection group W. In this paper, we establish Liouville-type theorems for the semilinear inequality − Δ k u ≥ | u | p in R N and the system of inequalities − Δ k u ≥ | v | p , − Δ k v ≥ | u | q in R N , where N ≥ 1 and p,q>1. To the best of our knowledge, this contribution is the first work dealing with Liouville-type results for nonlinear problems involving the Dunkl Laplacian.

Small Weights in Caccioppoli’s Inequality and Applications to Liouville-Type Theorems for Nonstandard Problems

Small Weights in Caccioppoli’s Inequality and Applications to Liouville-Type Theorems for Nonstandard Problems

Liouville type theorems for generalized P-harmonic maps

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature.

Improved energy decay estimate for Dir-stationary Q-valued functions and its applications

In this paper, we establish an improved decay estimate for the Dirichlet energy of Dir-stationary Q-valued functions. As a direct application of this estimate, we derive a Liouville-type theorem for bounded Dir-stationary Q-valued functions defined on Rm. Additionally, in an attempt to establish the continuity of Dir-stationary Q-valued functions, we confirm that such functions exhibit the Lebesgue property at every point within their domain.

Fully nonlinear dead-core systems

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system and obtain higher sharp regularity across the free boundary. Additionally, we prove geometric measure estimates and obtain coincidence of the free boundaries. We also derive Liouville type theorems for entire solutions. These results are new even for linear systems.

On Liouville type results for the stationary MHD in R3

This paper is concerned with the Liouville type theorems for the steady incompressible magnetohydrodynamics (MHD) equations. We establish that the solution to the steady MHD equations is identically zero under the integrability assumptions on (v, b). We show that, in particular, a combination of a strong integrability condition on the velocity of a fluid and a weak integrability condition on the magnetic field gives a sufficient condition on the Liouville type theorems. Furthermore, we show that the combination of the growth condition of the potential for the fluid velocity and the integrability condition for the magnetic field leads to the triviality of the solution.

A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system{−Δuj+λjuj=μjuj3+∑i≠jβijui2ujin B1,uj∈H0,r1(B1) for j=1,⋯,N. Here, λj,μj>0 and βij=βji are constants for i,j=1,⋯,N and i≠j. B1 denotes the unit ball in the Euclidean space R3 centred at the origin. For any P1,⋯,PN∈N, we prove the uniqueness of the radial solution (u1,⋯,uj) with uj changes its sign exactly Pj times for any j=1,⋯,N in the following case: λj≥1 and |βij| are small for i,j=1,⋯,N and i≠j. New Liouville-type theorems and boundedness results are established for this purpose.

Liouville-type theorems for the Taylor–Couette–Poiseuille flow of the stationary Navier–Stokes equations

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$ -norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$ -norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

An Obata-type formula and the Liouville-type theorem for a class of K-Hessian equations on the sphere

In this paper, we study a class of k k -Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts.