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

Abstract

We consider convex potentials $W:\mathbb{R}\to [0,\infty )$ vanishing at $0$ and growing sufficiently fast at $\pm \infty$. Given any open set $\Omega \subset {\mathbb{R}}^n$ with Lipschitz and compact boundary, we prove the existence and uniqueness of a solution of $\Delta u=W^{\prime }(u)$ 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 $A^{-}$.

10 pages