# 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…

## Figures from this paper

## 7 Citations

Classification of radial solutions for elliptic systems driven by the $k$-Hessian operator.

- Mathematics
- 2020

We are concerned with non-constant positive radial solutions of the system $$ \left\{ \begin{aligned} S_k(D^2 u)&=|\nabla u|^{m} v^{p}&&\quad\mbox{ in }\Omega,\\ S_k(D^2 v)&=|\nabla u|^{q} v^{s}…

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

- Mathematics
- 2020

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-\Delta)^{\frac{\alpha}{2}} u+\lambda\, \Delta u \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$,…

A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

- Mathematics
- 2019

Abstract We consider the elliptic equation - Δ u = u q | ∇ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} for any p > 2 {p>2} and q > 0 {q>0} . We prove a Liouville-type theorem,…

Asymptotic behavior of blowing-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

- Mathematics
- 2021

In this paper, we deal with the following quasilinear elliptic system involving gradient terms in the form: { ∆pu = v |∇u| in Ω ∆pv = v β |∇u| in Ω, where Ω ⊂ R (N ≥ 2) is either equal to R or equal…

Solvability of second-order uniformly elliptic inequalities involving demicontinuous $$\psi $$-dissipative operators and applications to generalized population models

- Mathematics
- 2021

Based on a new developed theory for variational inequalities, the purpose of this article is to investigate existence and uniqueness of nonzero positive weak solutions for a class of general…

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

- Mathematics
- 2022

. In this article we study local and global properties of positive solutions of − ∆ m u = | u | p − 1 u + M |∇ u | q in a domain Ω of R N , with m > 1, p, q > 0 and M ∈ R . Following some ideas used…

## References

SHOWING 1-10 OF 25 REFERENCES

Nonexistence results and estimates for some nonlinear elliptic problems

- Mathematics
- 2001

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(…

Classification of Radial Solutions for Semilinear Elliptic Systems with Nonlinear Gradient Terms

- Mathematics
- 2015

A new dynamical approach of Emden-Fowler equations and systems

- Mathematics
- 2010

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

- Mathematics
- 2013

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

- Mathematics
- 2016

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

- MathematicsSIAM J. Math. Anal.
- 2005

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

- Mathematics
- 2000

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…

Quasilinear elliptic systems in RN with multipower forcing terms depending on the gradient

- Mathematics
- 2013

Existence and a Priori Estimates for Positive Solutions of p-Laplace Systems

- Mathematics
- 2002

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

- Mathematics, Computer Science
- 2011