Existence of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity
Résumé
The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017]classified the behaviour near zero for all positive solutions of the perturbedelliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for$n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For$q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. thatthe positive solutions with a non-removable singularity at $0$ could exhibit upto three different singular profiles, although their existence was left open.In the present paper, we settle this question for all three singular profilesin the maximal possible range. As an important novelty for $\mu>0$, we provethat for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely manypositive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to\mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach.Moreover, we show that there exists a positive singular solution with$\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in(2^\star-2,2^\star-1)$.
Origine : Fichiers produits par l'(les) auteur(s)
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