The Boundary Blow-up Behavior For Quasilinear Elliptic Equations

By the Karamata regular variation theory, perturbation method and construction of comparison functions, we establish the boundary blow-up behavior of solutions for quasilinear elliptic equations.First, when nonlinear term/(u) may be rapidly varying at infinity, weight function b(x)∈Cα(Ω) may be vanishing on the boundary and rapidly varying near the boundary, we get first expansion of large solution for△pu=b(x)f(u), x∈Ω, u|(?)Ω=∞. Further, when f(s)=sm±f1(s)(where m> p-1, and f1is normalized regularly varying at infinity with index m1∈(0,m)) for sufficiently large s, we show the influence of mean curvature to the blow-up rate of large solution, that is, the second expansion of large solution near the boundary.Second, when nonlinear term f(u) is normalized regularly varying at infinity with index m (m> p-1), we show the influence of mean curvature to large solution. Here the weight function b(x) which may be vanishing on the boundary is non-negative and nontrivial.