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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References:

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