Related papers: Liouville-type theorems with finite Morse index fo…

We deal with one dimensional $p$-Laplace equation of the form $$ u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, $$ under Dirichlet boundary condition, where $p>2$ and $f\colon [0,l]\times \mathbb{R}\to \mathbb{R}$ is a continuous…

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…

The goal of this note is to consider Liouville type theorem for p-Laplacian type operators. In particular guided by the Laplacian case one establishes analogous results for the p-Laplacian and operators of this type.

In this paper, we study the Liouville-type property for smooth solutions to the steady 3D tropical climate model. We prove that if a smooth solution $(u,v,\theta)$ satisfies $u \in L^3 (\mathbb{R}^3)$, $v \in L^2 (\mathbb{R}^3)$, and…

The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of…

We are concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the…

We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is…

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix…

We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative $L^q$-viscosity solutions of the equation \begin{equation*} -\mathcal{M}_{\lambda, \Lambda}^{\pm}(D^2u)\pm |Du|^p=0, x\in…

In this paper we study the Liouville type properties for solutions to the steady incompressible Navier-Stoks equations in $\mathbf{R}^{3}$. It is shown that any solution to the steady Navier-Stokes equations in $\mathbf{R}^{3}$ with finite…

This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new…

In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic…

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator $$ L= \frac{1}{2}\text{ tr} (QD^2 ) + \langle Ax, D \rangle = \frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in R^N, $$ all (globally)…

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral…

Let $u$ be a solution of $\Delta u=Vu$ on $\mathbb{R}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $$\int_{r_j\leq|x|\leq r_j+1}|u(x)|^2dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow+\infty$,…

In this paper, we study the equation $\mathcal{L} u=0$ in $\mathbb{R}^N$, where $\mathcal{L}$ belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this…

In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted…

In this paper, we investigate the three dimensional stationary compressible Navier-Stokes equations, and obtain Liouville type theorems if a smooth solution $(\rho, \mathbf{u})$ satisfies some suitable conditions. In particular, our results…

In this paper, first we study carefully the positive solutions to $\Delta u+\lambda_{1}u\ln u +\lambda_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as…

We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ \Omega\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by…