Related papers: Higher-dimensional solutions for a nonuniformly el…
In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-\Delta)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with…
We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on…
We investigate the nonexistence and existence of nontrivial positive solutions to $\Delta_m u+u^p|\nabla u|^q\leq0$ on noncompact geodesically complete Riemannian manifolds, where $m>1$, and $(p,q)\in \mathbb{R}^2$. According to…
In this paper we establish gradient estimates for positive solutions to the nonlinear elliptic equation $$\Delta_{V}u^{m}+\mu(x)u+p(x)u^{\alpha}=0 , \quad m>1$$on any smooth metric measure space whose $k$-Bakry-\'{E}mery curvature is…
The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.
Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and…
We study Liouville theorems for the following polyharmonic H\'{e}non-Lane-Emden system \begin{eqnarray*} \left\{\begin{array}{lcl} (-\Delta)^m u&=& |x|^{a}v^p \ \ \text{in}\ \ \mathbb{R}^n,\\ (-\Delta)^m v&=& |x|^{b}u^q \ \ \text{in}\ \…
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…
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive…
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…
In \cite{LWZ}, we establish Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in…
In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u(y)=\intpr \frac{…
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum…
In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form…
In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ \Omega, \end{equation*}where $\Omega$…
We prove the existence of a family of non-trivial solutions of the Liouville equation in dimensions two and four with infinite volume. These solutions are perturbations of a finite-volume solution of the same equation in one dimension less.…
We study nonnegative solutions to the following Hardy-H\'enon type equations involving higher order fractional Laplacians $$ (-\Delta)^\sigma u = |x|^{-\alpha}u^{p} ~~~~~~ \mbox{in} ~ \mathbb{R}^n \backslash \{0\} $$ with a possible…
The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$…
We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex…
In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold \((M^n,g)\) with nonnegative Ricci curvature, namely \[-\Delta u =…