Singular solutions of some elliptic equations involving mixed absorption-reaction

Abstract

<p style='text-indent:20px;'>We study properties of nonnegative functions satisfying (E)<inline-formula><tex-math id="M1">\begin{document}$ \;-{\Delta} u+u^p-M|\nabla u|^q = 0 $\end{document}</tex-math></inline-formula> in a domain of <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ M&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 1&lt;q&lt;p $\end{document}</tex-math></inline-formula>. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q $\end{document}</tex-math></inline-formula> and in particular according to the sign of <inline-formula><tex-math id="M8">\begin{document}$ q-\frac{2p}{p+1} $\end{document}</tex-math></inline-formula>, and when <inline-formula><tex-math id="M9">\begin{document}$ q = \frac{2p}{p+1} $\end{document}</tex-math></inline-formula>, also on the value of the parameter <inline-formula><tex-math id="M10">\begin{document}$ M $\end{document}</tex-math></inline-formula> which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).</p>