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Let $\Omega$ be a smooth bounded domain in ${R}^N$. We prove general uniqueness results for equations of the form $- \Delta u = au - b(x) f(u)$ in $\Omega$, subject to $u = \infty$ on $\partial \Omega$. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow-up behavior of $u(x)$ with $f(u)$ and $b(x)$, where $b \equiv 0$ on $\partial \Omega$ and a certain ratio involving $b$ is bounded near $\partial \Omega$. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov.

The aim of this paper is to show the existence of solutions with an arbitrarily large number of bubbles for the slightly super-critical elliptic problem $-\Delta u=u^{{(N+2)/(N-2)} +\ve }$ in $\Omega$, subject to the conditions that $u>0$ in $\Omega$, and $u=0$ on $\partial \Omega$, where $\ve >0$ is a small parameter and $\Omega \subset \RR^N$ is a bounded domain with certain symmetries, for instance an annulus or a torus in $\RR^3$.

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General uniqueness results and variation speed for blow-up solutions of elliptic equations

MULTI-BUBBLE SOLUTIONS FOR SLIGHTLY SUPER-CRITICAL ELLIPTIC PROBLEMS IN DOMAINS WITH SYMMETRIES

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General uniqueness results and variation speed for blow-up solutions of elliptic equations

MULTI-BUBBLE SOLUTIONS FOR SLIGHTLY SUPER-CRITICAL ELLIPTIC PROBLEMS IN DOMAINS WITH SYMMETRIES