Explosive solutions of quasilinear elliptic equations: Existence and uniqueness.

*(English)*Zbl 0793.35028This paper deals with the quasilinear elliptic equation

$$-\text{div}\left(Q\left(\left|\nabla u\right|\right)\nabla u\right)+\lambda \beta \left(u\right)=f\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega}\subset {\mathbb{R}}^{N},\phantom{\rule{4pt}{0ex}}N>1;$$

more precisely, existence and uniqueness of local solutions satisfying

$$u\left(x\right)\to \infty \phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}\text{dist}(x,\partial {\Omega})\to 0$$

and other properties are the main goals here. These kinds of functions are called explosive solutions. No behaviour at the boundary to be prescribed is a priori imposed. However, we are going to show that, under an adequate strong interior structure condition on the equation, explosive behaviour near $\partial {\Omega}$ cannot be arbitrary. In fact, there exists a unique such singular character governed by a uniform rate of explosion depending only on the terms $Q$, $\lambda $, $\beta $ and $f$.