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

## TL;DR

This paper establishes the existence and uniqueness of solutions to a class of semilinear PDEs with convex potentials that blow up at the boundary of a domain, including cases with boundary subsets where the solution diverges to infinity.

## Contribution

It proves the existence and uniqueness of boundary blow-up solutions for semilinear PDEs with convex potentials on Lipschitz domains, including boundary conditions with multiple disjoint subsets.

## Key findings

- Unique solutions with boundary blow-up behavior are guaranteed.
- Solutions exist for convex potentials growing fast at infinity.
- The results cover cases with multiple boundary subsets with specified blow-up conditions.

## 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 $A^-$.