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

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 $q$.