Asymptotic behavior of large solutions to H-systems with perturbations

Abstract

Let Ω⊂ R 2 be a bounded smooth domain. In this paper, we study the existence and the asymptotic behavior of solutions to the Dirichlet problem of H-systems with perturbations: (P ε) Δu=2u x 1 ∧u x 2 −εu in Ω, u| ∂Ω=0, where u=(u 1,u 2,u 3)∈H 0 1(Ω; R 3) and ε∈ R . We prove that for ε∈(0,λ 1) (λ 1 denotes the 1st eigenvalue of − Δ acting on H 0 1(Ω; R 3) ), ( P ε ) admits at least one solution u ̄ ε≢0 , which blows up at exactly one point in Ω as ε→0. We also characterize the location of blow up point as the minimum point of a certain function defined on Ω, and give the exact blow up rate of || ∇ u ̄ ε|| L ∞(Ω) as ε→0.