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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 $$-\text{div} \biggl( Q \bigl( \vert \nabla u \vert \bigr) \nabla u \biggr)+\lambda \beta (u)=f \quad \text{in } \Omega \subset \bbfR\sp N,\ N>1;$$ more precisely, existence and uniqueness of local solutions satisfying $$u(x) \to \infty \quad \text{as dist} (x,\partial \Omega) \to 0$$ 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 $f$.

35J60Nonlinear elliptic equations
35B40Asymptotic behavior of solutions of PDE
Full Text: DOI
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