Existence of single-peaked solution of a semilinear elliptic problem

We discussed the Dirichlet problem of semilinear elliptic equation (P _β,ε):β ²Δu=u ^p +εu,u>0, in Ω;u=0, on ∂Ω, where Ω⊂R ^N (N≥4) is smooth and bounded domain, ,β,ε>0. We have proved that there exist positiveε ₀ andε ₁, such that when 0⩽ε⩽ε ₀,β>√ε ₁, (P _β,ε) has a single-peaked solutionu _β,ε; furthermore, |∇u _β0|²⇀0 in the sense of measure as ε→0 and β→0. Foundation item: Supported by the National Natural Science Foundation of China, Foundation for Fundamental Sciences of Nanchang University and Hua-chen Foundation. Biography: Zhong Xin-hua (1972-), famale, Ph. D. candidate, lecture, research interest: Partial Differiential Equation. The author's permanent address is: Department of Mathematics, Nanchang University, Nanchang 330047, China.