Existence And Multiplicity Of Solutions For Two Classes Of Nonlocal Problems

First,we consider a class of nonlocal problems where a>0,b>0,Ω(?)RN is a bounded open set,g∈ H-1(Ω)\{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(Ω)\{0}.Then there existsλ*>0 such that(1)The problem has at least three solutions if λ ∈(0,λ*),(2)The problem has at least two solutions if λ =λ*,(3)The problem has at least one solution if λ>λ*.Then we consider another class of nonlocal problems where a>0,b>0,Ω(?)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 ∈ Ω.(f2)There exist h ∈ L2*/2*-1(Ω),1 ≤ p<2*and g ∈ L2*/2*-p(Ω)such that|f(x,t)| ≤ h(x)+g(x)|t|P-1 for every t ∈ R and a.e.x∈ Ω.(fs)f(x,-t)=-f(X,t)for every t ∈ R and a.e.x∈ Ω.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).