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

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 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.