Existence And Multiplicity Of Solutions For Two Classes Of Nonlocal Problems

First,we consider a class of nonlocal problems where a>0,b>0,?(?)RN is a bounded open set,g? H-1(?)\{0}.The Ekeland's variational principle and the mountain pass lemma are applied to prove the following theorem.Theorem 0.1.Assume a,b>0,and g? H-1(?)\{0}.Then there exists?*>0 such that(1)The problem has at least three solutions if ? ?(0,?*),(2)The problem has at least two solutions if ? =?*,(3)The problem has at least one solution if ?>?*.Then we consider another class of nonlocal problems where a>0,b>0,?(?)C RN is a bounded open set.Suppose that f satisfies the following conditions.(f1)f:Q x R ?R is a Caratheodory function and f(x,t)t>0 for every t ? R\{0} and a.e.x ? ?.(f2)There exist h ? L2*/2*-1(?),1 ? p<2*and g ? L2*/2*-p(?)such that|f(x,t)| ? h(x)+g(x)|t|P-1 for every t ? R and a.e.x? ?.(fs)f(x,-t)=-f(X,t)for every t ? R and a.e.x? ?.The equivariant link theorem is applied to prove that the following theorem.Theorem 0.2.Assume that a>0,b>0 and(f1),(f2)and(f3)hold.Then problem(0.4)has infinitely many solutions who's energy increasingly tends to a2/(4b).