We examine the general weighted Lane-Emden system $-Δ u = ρ(x)v^p, -Δ v= ρ(x)u^θ, u,v>0 \;\mbox{in }\;\mathbb{R}^N$ where $1 <p≤qθ$ and $ρ: \mathbb{R}^N \to \mathbb{R}$ is a radial continuous function satisfying $ρ(x)≥q A(1+|x|^2)^{\frac{α}{2}}$ in $\mathbb{R}^N$ for some $α≥q 0$ and $A>0$. We prove some Liouville type results for stable solution and improve the previous works [2, 9, 12]. In particular, we establish a new comparison property (see Proposition 1 below) which is crucial to handle the case $1 < p ≤q \frac{4}{3}$. Our results can be applied also to the weighted Lane-Emden equation $-Δ u = ρ(x)u^p$ in $\mathbb{R}^N$.

We consider the Lane–Emden system , in , and we prove the nonexistence of smooth positive solutions which are stable outside a compact set, for any under the Sobolev hyperbola.

We examine the following degenerate elliptic system: $ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v>0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!>\!0. $ We prove that the system has no stable solution provided $ p, \theta >0 $ and $ N_s: = N_1+(1+s)N_2< 2 + \alpha + \beta, $ where $ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $ This result is an extension of some results in [ 15 ]. In particular, we establish a new integral estimate for $ u $ and $ v $ (see Proposition 1.1), which is crucial to deal with the case $ 0 < p < 1. $

In this work, we investigate the stabilization problem for coupled biharmonic Schrödinger equations with fractional internal damping. First, we establish that the system is well-posed using the semigroup theory of linear operators. Second, we demonstrate that the system is strongly stable. Finally, we prove a polynomial decay rate using multiplier techniques combined with the frequency domain method, and we show the lack of exponential decay of the energy in a specific case.