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Pré-Publication, Document De Travail Année : 2018

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

Résumé

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 $q$.
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Dates et versions

hal-01651511 , version 1 (30-11-2017)
hal-01651511 , version 2 (05-01-2018)
hal-01651511 , version 3 (15-12-2018)

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Marie-Françoise Bidaut-Veron, Marta Garcia-Huidobro, Laurent Veron. Estimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradient. 2018. ⟨hal-01651511v2⟩
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