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We consider convex potentials $W:\R\to [0,\infty)$ vanishing at $0$ andgrowing sufficiently fast at $\pm\infty$. Given any open set$\Omega\subset\R^n$ with Lipschitz and compact boundary, we prove the existenceand uniqueness of a solution of $\Delta u= W'(u)$ in $\Omega$, such that$u=+\infty$ or $u=-\infty$ on $\partial \Omega$. Moreover, if $\partial \Omega$is the union of two disjoint compact subsets $A^+$ and $A^-$, there also existsa unique solution satisfying $u=+\infty$ on $A^+$ and $u=-\infty$ on $A^-$.

Blowing up solutions of semilinear P.D.E. with convex potentials