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DCDS

Let $b(x)$ be a positive function in a bounded smooth domain
$\Omega\subset R^N$, and let $f(t)$ be a positive non decreasing
function on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$.
We investigate boundary blow-up solutions of the equation $\Delta
u=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$
approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity,
we find a second order approximation of the solution $u(x)$ as $x$
goes to $\partial\Omega$.

We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.

We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.

PROC

We investigate blow-up solutions of the equation $\Deltau$ = $f(u)$ in a bounded smooth domain $\Omega \subset R^N$. Under appropriate growth conditions on $f(t)$ as $t$ goes to infinity we show how the mean curvature of the boundary $\partial\Omega$ appears in the second order term of the asymptotic expansion of the solution $u(x)$ as $x$ goes to $\partial\Omega$.

PROC

In the rst part of this paper we discuss a minimization problem where symmetry breaking arise. Consider the principal eigenvalue for the problem -$\Deltau = \lambdaxFu$ in the ball $B_(a+2) \subset\mathbb{R}^N$, where $N\>= 2$ and $F$ varies in the annulus $B_(a+2)\\B_a$, keeping a xed measure. We prove that, if a is large enough, the minimum of the corresponding principal eigenvalue is attained in a subset which is not symmetric. In the second part of the paper we show that the principal eigenvalue of an equation with an indenite weight in a general bounded domain $\Omega$ can be approximated by a functional related to the energy integral of a suitable nonlinear equation; furthermore, we show that a solution of this nonlinear equation approximates, in the $H^1(\Omega)$ norm, the principal eigenfunction of our problem.

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