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 $1\<p+q$, $p\ge 0$, $0\leq q\< 2$. 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 $u(x)\le c(\mathrm{d}ist(x,\partial \Omega {)}^{\frac{q-2}{p+q-1}}$ 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 $q$.