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

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