# Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms

@article{Chang2020LiouvilletypeTA, title={Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms}, author={Caihong Chang and Bei Hu and Zhengce Zhang}, journal={arXiv: Analysis of PDEs}, year={2020} }

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show the Liouville-type theorems for positive weak solutions of the equation involving the $m$-Laplacian operator \begin{equation*} -\Delta_{m}u=u^q|\nabla u|^p\ \ \ \mathrm{in}\ \mathbb{R}^N, \end{equation*} where $N\geq1$, $m>1$ and $p,q\geq0$. This paper mainly adopts the technique of Bernstein gradient estimates to study from two cases: $p=m$ and…

## One Citation

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- 2021

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is F (x,Du) ≥ Hi(x, u,Du) in R , where Hi…

## References

SHOWING 1-10 OF 82 REFERENCES

Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

- Mathematics
- 2016

In this paper we consider positive supersolutions of the elliptic equation $-\triangle u =
f(u) |\nabla u|^q$, posed in exterior domains of $\mathbb{R}^N$ ($N\ge 2$), where $f$ is
continuous in…

A priori estimates and existence for quasi-linear elliptic equations

- Mathematics
- 2008

AbstractWe study the boundary value problem of quasi-linear elliptic equation
$$\begin{array}{rl} {\rm div}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) = 0 &\quad {\rm in}\, \Omega,\\ u = 0 &\quad…

Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient

- MathematicsDuke Mathematical Journal
- 2019

We study local and global properties of positive solutions of $-{\Delta}u=u^p]{\left |{\nabla u}\right |}^q$ in a domain ${\Omega}$ of ${\mathbb R}^N$, in the range $1<p+q$, $p\geq 0$, $0\leq q< 2$.…

Asymptotic behavior of global solutions to a class of heat equations with gradient nonlinearity

- Physics
- 2020

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: \begin{document}$ u_t = u_{xx}+x^m|u_x|^p, \ \ t>0, \ \ 0 where \begin{document}$ p>m+2…

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,…

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

- Mathematics
- 2017

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation $${u_t} - div\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) = - {\left| u…

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

- Physics, Mathematics
- 2019

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive…

Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations

- Mathematics
- 2008

If $h$ is a nondecreasing real valued function and $0\leq q\leq 2$, we analyse the boundary behaviour of the gradient of any solution $u$ of $-\Delta u+h(u)+\abs {\nabla u}^q=f$ in a smooth…

Asymptotic behavior of solutions for a free boundary problem with a nonlinear gradient absorption

- MathematicsCalculus of Variations and Partial Differential Equations
- 2019

This paper deals with the free boundary problem for a parabolic equation, $$u_t-u_{xx}=u^{p}-\lambda |u_x|^{q}$$ut-uxx=up-λ|ux|q, $$t>0$$t>0, $$01$$p,q>1. It is well known that global existence or…

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(…