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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 \[ -\text{div} \biggl( Q \bigl( | \nabla u | \bigr) \nabla u \biggr)+\lambda \beta (u)=f \quad \text{in } \Omega \subset \mathbb{R}^ N,\;N>1; \] 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 \(f\).

MSC:

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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