相关论文: The H\'enon-Lane-Emden system: a sharp nonexistenc…
In this paper, we address the solvability of the critical Lane-Emden system \[\begin{cases} -\Delta u=|v|^{p-1}v &\mbox{in } \Omega_\epsilon,\\ -\Delta v=|u|^{q-1}u &\mbox{in } \Omega_\epsilon,\\ u=v=0 &\mbox{on } \partial \Omega_\epsilon,…
We investigate the H\'enon-Lane-Emden system defined by $- \Delta u=|x|^a |v|^{p-1}v$ and $- \Delta v=|x|^b |u|^{q-1}u$ in $\mathbb{R}^N \!\setminus\! \{0\}$. We begin by establishing a general Liouville-type theorem for the subcritical…
We prove a sharp local existence result for the Schr\"odinger-Korteweg-de Vries system with initial data in $H^k(\mathbb{R})\times H^s(\mathbb{R})$. The proof is based on the concept of \textit{integrated-by-parts strong solution}, which…
We prove the existence of nonradial solutions for the H\'enon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $\alpha$. For sign-changing solutions, the case $\alpha=0$ -- Lane-Emden equation…
In this paper, we first establish decay estimates for the fractional and higher-order fractional H\'enon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the…
We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system structure and uses Carleman estimates. We apply this result to obtain some…
We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The…
We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to…
This paper and [17] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy--H\'enon equation $(-\Delta)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$ where $0 < s < 1$, $\ell > -2s$,…
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -\Delta u= h(x,u)\quad \ &{\rm in}\ \Omega,\\[1.5mm] \phantom{ -\Delta }…
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…
We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial…
The aim of this paper is to establish strong convergence theorems for a strongly relatively nonexpansive sequence in a smooth and uniformly convex Banach space. Then we employ our results to approximate solutions of the zero point problem…
Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{…
We study positive solutions of the superlinear Lane-Emden inequality \(-\Delta u\ge \sigma u^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the…
We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…
We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy…
In this work, we study the existence and nonexistence of solution for strongly coupled elliptic systems to m-parameters.
The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity…
We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and…