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In this paper,we stndy the asymptotic behavior of solutions for semilinear elliptic sys tems with a nearly critical exponentχ∈Ω,u=v＝0,x∈(?)Ω,whereΩ(?)Rn is a ball centere at the origin.ε>0,α,β> 0,pε=p－(p＋1)ε,qε－q-(q+1)s,p,q>1,1/p+1+1/q+1=(N-2)/2.By the concentration compactness principle of P.L.Lions and th blow—up ways of Gidas-Spruck,we proved the fowing two t heorems.Theorenms 1.1 Let 0<α<pN,β>0，p，q>1.(uε,vε)is the groun solutionion of the equation(0.1)，in the scese of subsequence,whenε→0,there exist,s x0 ∈(?)Ωsatisfying，(i)in the sense of measure，|(ii)in t he sense of measure，| whreeμ>0，v>0 satisfyμ≥Sp.qVq(p+1),δχis the Dirac measure of the point of:Theorems 1.2 Let 0<α<pN，β＞>0,p,q>1 and max{2(p+1)/(pq-1),2(q+1)/(pq-1)}>N-3,(uε,vε) is the ground solution of the equation(0.1)，xε∈Ωsatisfyμε=Mε－N/(p+1),Mε＝uε(xε)= maxxn∈uε(x),whenε→0,we can conclude Mε→∞and(i)whenεis small enough,Xεis unique，and whenε→0 we call get dist(xε，6Ω)→0,dist(xε,(?)Ω)→∞; where Uμε,xε(x)=με－(N)/p+1∪(x-xε/με),and (U,V)is the groun(I sohnion of the equal.ion—△u=vp,-△v=uq, x∈RNThis paper includes three chapters.In chapter 1,we present a simple research summary of the asymptotic behavior of solu-tions for semilinear elliptic systems with a nearly critical exponent,list the main theorems in this paper,theoremsl.1 and theorems1.2.In chapter 2,we list some preknwqledges and lemmas.In chapter 3,we present the process proved of theoremsl.1 and theoremsl.2.
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