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 semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.

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 $\RR^3$.