Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities

We extend our recent results on the classification of stable solutions of the equation −Δu=f(u)$-\Delta u=f\left(u\right)$ in R^N${\mathbb{R}}^N$, where f≥0$f\ge 0$ is a general convex, non-decreasing function.     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

Liouville theorems for non-local operators

The paper characterizes some classes of pseudo-differential operators for which there are (or there are not) non-constant bounded harmonic functions. Non-local perturbations of Ornstein-Uhlenbeck operators and operators with dissipative coefficients are considered. The methods used are probabilistic and based on the concept of absorption function and on a new extension of the Bismut-Elworthy-Li formula. The probabilistic interpretation of the Liouville theorem by means of absorption functions for general Markov processes is given as well.     

Liouville theorems for III-posed problems

The growth rate at infinity of non-constant solutions of certain abstract differential equations is studied making essential use of the convexity of the norm of the solution.     

Liouville theorems for quasi-harmonic functions

Let N be a compact Riemannian manifold. A self-similar solution for the heat flow is a harmonic map from to N (n≥3), which was also called a quasi-harmonic sphere (cf. Lin and Wang (1999) [1]). (Here is the Euclidean metric in .) It arises from the blow-up analysis of the heat flow at a singular point. When and without the energy constraint, we call this a quasi-harmonic function. In this paper, we prove that there is neither a nonconstant positive quasi-harmonic function nor a nonconstant quasi-harmonic function. However, for all 1≤p≤n/(n−2), there exists a nonconstant quasi-harmonic function in .     

Liouville theorems for harmonic maps

Summary We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R ^m ,g ₀) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at .Oblatum 28-XII-1990 & 11-II-1991Supported by NSF grant DMS-8610730     

Extensions of Liouville theorems

The method of deriving Liouville's theorem for subharmonic functions in the plane from the corresponding Hadamard three-circles theorem is extended to a more general and abstract setting. Two extensions of Liouville's theorem for vector-valued holomorphic functions of several complex variables are also mentioned.     

On solutions of a boundary value problem for the biharmonic equation

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.     

Liouville theorems for some nonlinear inequalities

We prove various Liouville theorems for integral and differential inequalities on the whole ℝR ^N . The main tools we use throughout this paper are representation formulae for linear inequalities, the nonlinear capacity method and the weak form of Harnack’s inequality. This paper is dedicated to Professor Stanislav I. Pohozaev with admiration and friendship     

The blow-up rate for positive solutions of indefinite parabolic problems and related Liouville type theorems

In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.