Large Solutions of Elliptic Equations with Strong Absorption

Abstract

We present some general results dealing with existence and uniqueness of solutions of −Δu + g(x, u) = 0 in a domain $\Omega \subset \mathbb{R}^N$, which satisfy $\lim _{dist\left( {x,\partial \Omega } \right) \to 0} u\left( x \right) = \infty$, where g is a continuous nonnegative function. We emphasize the links between the regularity of the boundary and the existence of such solutions. The cases $g\left( {x,r} \right) = \rho ^\alpha \left( x \right)r_ + ^q$ and $g\left( {x,r} \right) = \rho ^\alpha \left( x \right)e^{br}$ are thoroughly investigated.