Existence Of Multiple Solutions For Boundary Value Problems Of Second Order Subquadratic Equations

In this paper,we mainly study of the existence and multiplicity results of sublinear elliptic equation of second-order(?)via critical point theory arid Z2-index theory,here ??E R is parameter with N a(x)? N/L2(?2),N? 3 and W(x,u)?C1(?,R),The corresponding eigenvalue problem is(?)and the ? is eigenvalue with-?<?1<?2? ?3??…?n? 0<?An+1??n+2 ?…·<+?,the corresponding eigenspaces Nj = {uj}(j ? 1)have finite dimensions.The corresponding functional of the second elliptic equation is(?)The critical point on this functional are the weak solutions of the equation,these also are the classical solutions.We could study the multiplicity of function-nal critical point via index theory that in critical point theory.This paper consists of four chapters:The first chapter is the introduction,it mainly introduces the background of this paper,research progress and the main conclusion of this paper.The second chapter introduces the relevant knowledge of this paper.The third chapter introduces sublinear elliptic equation of second-order and study the lemmas in three situations:(i)?<?1;(?)?k<?<?k+1;(iii)?= ?k.The fourth chapter is key chapter that detailed proof for the main conclusion.For the situation(i),to prove that?(u)satisfies the condition(P.S),then using Clark theory to prove?(u)has multiple critical points;for the situation(ii)and situation(iii),first prove it has one norm ?·? is equivalent to norm ?·?0,thus for every sequence has bounded;then prove {um}(?)X has convergent subsequence,thus?(u)subject to condition(P.S),last prove ?(u)has multiple critical points via index theory.