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Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient. (arXiv:1711.11489v1 [math.AP])
来源于:arXiv
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 $1p+q$, $p\geq 0$, $0\leq q 2$. 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
$0\leq q1$. 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
$p+q1+\frac{4}{n-1}$. 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 查看全文>>