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              <p>This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE's as $\Delta u=f(u)$ where $f$ satisfies the so-called Keller-Osserman condition. We {\bleu characterize} existence of such solutions {\bleu for non-monotone $f$} . As an example, we construct an infinite family of boundary blow up solutions for the equation ${\bleu \Delta u=u^2(1+\cos u)}$ on a ball. We {\bleu prove} uniqueness {\bleu (on balls) when} $f$ is increasing and convex {\bleu in a} neighborhood of infinity and we discuss and perform some numerical computations to approximate such boundary blow-up solutions</p>
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