Asymptotic Behavior Of Solutions Of A Nonlinear Equation

In this paper, we study asymptotic behavior of solutions of a nonlinear func-tion as-Δu=λf(u) with Dirichlet conditions,here we suppose that the nonlinear term f(u) is asymptotically linear,we talk about the limit by estimating the first eigenfunction,hence we can obtain the growth of u.First,we estimate (?)I(φ1>ε) 1/φ1θby using of previous results and some skills.Then we consider the growth of u when f(t)=t+1/(t+1)θ,θ>2,that is the results of (?)(λ1-λ)α‖u(λ)‖L2(Ω).we will talk about it according to four cases ofα.At last we will consider general cases,that is f(t)-at=O((1/t)θ),θ2. we will get the growth of u through two steps,i.e. (?)(λ1-λ)α‖u(λ)‖L2(Ω).