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Explosive solutions of quasilinear elliptic equations: Existence and uniqueness. (English) Zbl 0793.35028

This paper deals with the quasilinear elliptic equation

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

more precisely, existence and uniqueness of local solutions satisfying

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

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 $f$.

##### MSC:
 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions of PDE