Existence Of Multiple Radial Solutions For Robin Problems Of Elliptic Partial Differential

This paper is mainly composed of two chapters:In chapter1we discuss Robin boundary value problems on the unit ball一△υ=f(υ),χ∈B1；υ+βan|au=0,χ∈a.B1.Chapter2studies the boundary value problem with Robin outer boundary situation in annular region一△υ=f(υ),χ∈B1\B|E；an|au=0,x∈aBε；υ+βan|au=0,x∈aBl.For each category of problems we obtain via the Leggett-Williams three solutions theorem the existence of at least three nonnegative radial solutions.Here the conclusions of chapter1and chapter2.Consider the following problem where f∈C(R+,R+),R+=[0,∞).β≥0,B1={x∈RN:|x|<1｝,aBl={x∈RN:|x|=1｝,N≥3,an|au denotes exterior normal derivative.We have the following theorem:Theorem1.1Assume there are0<d<a<c satisfying a/co<2Nc/(1+2β),such that the nonlinear term f satisfies:(H1)f(υ)<2Nd/(1+2β),υ∈[0,d]；(H2)f(υ)>a/co,υ∈[a,c]；(H3)f(υ)≤2Nc/(1+2β),μ∈[0,c], then the problem(1)has at least three nonnegative radial solutions,whereConsider the following problem where f∈C(R+,R+),β≥0,Bε={x∈RN:|x|≤ε},aBε={x∈RN:|X|=ε｝,ε∈(0,1/2),N≥3,an|au denotes exterior normal derivative.We have the following theorem:Theorem2.1Assume there are0<d<a<b≤c,such that the following conditions are established:(H4)a/c1<c/c2,σb>a；(H5)f(υ)<d/c2,υ∈[O,d]；(H6)f(u)>a/c1,υ∈[a,b]；(H7)f(υ)≤c/c2,υ∈[O,c], whereσ=2-N|1-2-N|1(2|1)2-N+β|2-N|1-2-N|1ε2-N+β,c1=σιε1/2G2(s,s)ds,c2=ιε1/2G2(s,s)ds,then the prob-lem(2)has at least three nonnegative radial solutions.