Related papers: A Liouville theorem for a fourth order H\'enon equ…

In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the…

This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-\Delta)^2 u=\lambda u+…

For $N\geq 4$, we let $\Omega$ to be a smooth bounded domain of $\mathbb{R}^N$, $\Gamma$ a smooth closed submanifold of $\Omega$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $\Omega$. We denote by…

In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: \begin{equation}\label{GPDE0} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\,\,…

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue…

Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation,…

In this paper, we investigate Liouville theorems for solutions to the anisotropic $p$-Laplace equation $$-\Delta_p^H u=-\operatorname{div}(a(\nabla u))=f(u),\quad\text{in }\mathbb{R}^n,$$ where the semilinear term $f$ may be positive,…

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator…

In this paper, we are concerned with the fractional and higher order H\'{e}non-Hardy type equations \begin{equation*} (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\,…

We give a complete classification of solutions bounded from above of the Liouville equation $$-\Delta u=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty}…

We prove the nonexistence of smooth stable solution to the biharmonic problem $\Delta^2 u= u^p$, $u>0$ in $\R^N$ for $1 < p < \infty$ and $N < 2(1 + x_0)$, where $x_0$ is the largest root of the following equation: $$x^4 -…

In this paper, we consider the weighted fourth order equation $$\Delta(|x|^{-\alpha}\Delta u)+\lambda \text{div}(|x|^{-\alpha-2}\nabla u)+\mu|x|^{-\alpha-4}u=|x|^\beta u^p\quad \text{in} \quad \mathbb{R}^n \backslash \{0\},$$ where $n\geq…

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-\Delta ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot \nu =0 \quad\,\,\, &{\rm{on}} \,\,…

In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schr\"odinger equation $$ iu_t-\mu\Delta u+\Delta^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $\mu\ge 0$. As an…

We use maximum principle to prove the Liouville theorem of the equation $\Delta U + b\cdot \nabla U + h U^{\alpha} = 0, U \geq 0, 0 < \alpha < \frac{n + 2}{n - 2}$ on the complete Riemannian manifold with non-negative Ricci tensor, which…

In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\Delta u=\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\partial…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial \Omega$ and $\Sigma$ be a closed submanifold contained on $\partial \Omega$ and containing $0$. We are interesting in the existence of…

We consider the Lane-Emden system $-\Delta u = v^p$, $-\Delta v= u^\theta$ in $\mathbb{R}^N$, and we prove the nonexistence of smooth positive solutions which are stable outside a compact set, for any $p, \theta > 0$ under the Sobolev…

Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order H\'enon equation $\Delta^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu…

For the equation (-\Delta u = | |x|-2 |^\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A…