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Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

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Abstract

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

$ - \varepsilon \Delta _p u = f(x,u)in\Omega ,$

where 1 < p s ∞, ɛ > 0 is a small parameter,

$f(x,u) = \left\{ \begin{gathered} \left| u \right|^{p - 2} u\left| {u - a(x)} \right|^{p - 2} (u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),ifp \geqslant 2 \hfill \\ \left| u \right|^{(2 - p)/(p - 1)} u(u - a(x))\left| {1 - u} \right|^{\omega - 1} (1 - u),if1 < p < 2, \hfill \\ \end{gathered} \right. $

where ω > 0, a(x) is a continuous function satisfying 0 < a(x) < 1 for x ∈ \(\bar \Omega \), Ω is a bounded smooth domain in ℝN. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.

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  1. Department of Mathematics, He’nan Normal University, Xinxiang, 453007, P. R. China

    Zong Ming Guo & Yao Yong Yan

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  1. Zong Ming Guo
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  2. Yao Yong Yan
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Correspondence to Zong Ming Guo.

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Supported by National Natural Science Foundation of China (Grant No. 10871060)

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Guo, Z.M., Yan, Y.Y. Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems. Acta. Math. Sin.-English Ser. 27, 2177–2190 (2011). https://doi.org/10.1007/s10114-011-9327-0

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  • DOI: https://doi.org/10.1007/s10114-011-9327-0

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