Quasilinear elliptic equations with boundary blow-up
This paper is devoted to the problem \(\triangle_pu(x)=f(u(x))\) in a bounded domain \(D\), \(u(x)\to\infty\) as \(x\to\partial D\), where \(\triangle_p\) is the \(p\)-Laplacian. The existence of a solution is proved and its asymptotic behaviour near the boundary is studied. As for the Laplacian it depends only on the nonlinearity and not on the geometry. The method relies on comparison functions and uses the maximum principle for \(p\)-Laplacians. The author also extends a method developed for the Laplacian by Bandle and Essen, for estimating the gradients to the more general operator. This method uses a scaling argument and applies to power nonlinearities.
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