Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

@article{Ghergu2019GlobalAB,
  title={Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids},
  author={Marius Ghergu and Jacques Giacomoni and Gurpreet Singh},
  journal={Nonlinearity},
  year={2019}
}
We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q} &&\quad\mbox{ in }\Omega, \end{aligned} \right. $$ where $\Omega\subset\R^N$ $(N\geq 2)$ is either a ball or the whole space, $1 0$, $\alpha\geq 0$, $0\leq \beta\leq m$ and $(p-1-\alpha)(p-1-\beta)-qm\neq 0$. We first classify all the positive radial solutions in case… 

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References

SHOWING 1-10 OF 25 REFERENCES
Nonexistence results and estimates for some nonlinear elliptic problems
AbstractHere we study the local or global behaviour of the solutions of elliptic inequalities involving quasilinear operators of the type $$L_{\mathcal{A}^u } = - div\left[ {\mathcal{A}\left(
A new dynamical approach of Emden-Fowler equations and systems
We give a new approach on general systems of the form \[ (G)\left\{ \begin{array} [c]{c}% -\Delta_{p}u=\operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon_{1}\left\vert
Eternal solutions to a singular diffusion equation with critical gradient absorption
The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular
Coercive elliptic systems with gradient terms
Abstract In this paper we give a classification of positive radial solutions of the following system: Δ ⁢ u = v m , Δ ⁢ v = h ⁢ ( | x | ) ⁢ g ⁢ ( u ) ⁢ f ⁢ ( | ∇ ⁡ u | ) , $\Delta u=v^{m},\quad\Delta
Large Solutions for a System of Elliptic Equations Arising from Fluid Dynamics
TLDR
This paper proves that the elliptic system (0.1) has a unique radially symmetric and nonnegative large solution with v(0) = 0 (obviously, v is determined only up to an additive constant) and study the asymptotic behavior of these solutions near the boundary of Omega and determine the exact blow-up rates.
Local behaviour of the solutions of a class of nonlinear elliptic systems
Here we study the behaviour near a punctual singularity of the positive solutions of semilinear elliptic systems in R (N 3) given by u+ jxj uv = 0; v + jxj uv = 0; (where a; b; p; q; s; t 2 R , p; q
Existence and a Priori Estimates for Positive Solutions of p-Laplace Systems
We use continuation and moving hyperplane methods to prove some existence and a priori estimates for p-Laplace systems of the form−Δp1u=f(∣v∣) in Ω,u=0 on ∂Ω,−Δp2v=g(∣u∣) in Ω,v=0 on ∂Ω, where
Nonexistence of nonnegative solutions of elliptic systems of divergence type
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2
3
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