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

TL;DR

This paper investigates positive solutions of a nonlinear elliptic PDE involving the function and its gradient, establishing local and global properties, nonexistence results, and conditions for solutions to be constant or have specific forms.

Contribution

It introduces new bounds and nonexistence results for solutions, extending previous work and analyzing solutions with specific spherical symmetry forms.

Findings

Proved local Harnack inequality for solutions.

Established nonexistence in certain unbounded domains.

Identified conditions for solutions to be constant or radially symmetric.

Abstract

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 $1\<p+q$, $p\geq 0$, $0\leq q\< 2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\mathbb R}^N$ when $p(N-2)+q(N-1) \<N$ or in an exterior domain if $p(N-2)+q(N-1)\<N$ and $0\leq q\<1$. Using a direct Bernstein method we obtain a first range of values of $p$ and $q$ in which $u(x)\leq c({\mathrm dist\,}(x,\partial\Omega)^{\frac{q-2}{p+q-1}}$ This holds in particular if $p+q\<1+\frac{4}{n-1}$. Using an integral Bernstein method we obtain a wider range of values of $p$ and $q$ in which all the global solutions are constants. Our result contains Gidas and Spruck nonexistence result as a particular case. We also study solutions under the form $u(x)=r^{\frac{q-2}{p+q-1}}\omega(\sigma)$. We prove existence, nonexistence and rigidity of the spherical component $\omega$ in some range of values of $N$, $p$ and $q$.