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

Abstract

We consider convex potentials $W:\R\to [0,\infty)$ vanishing at $0$ and growing sufficiently fast at $\pm\infty$. Given any open set $\Omega\subset\R^n$ with Lipschitz and compact boundary, we prove the existence and 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 exists a unique solution satisfying $u=+\infty$ on $A^+$ and $u=-\infty$ on $A^-$.