A Class Of Fourth-order Navier Boundary Value Problems Of Existence And Multiplicity Of Solutions

In this paper, we study the existence and multiplicity of weak solutions for a class of fourth-order Navier boundary value condition by using the least action principle, the Ekeland variational principle and the mountain pass theorem.Consider the following fourth-order Navier boundary value problem where Δ2denotes the biharmonic operator, Ω(?)RN(N>4) is a smooth bounded domain, a∈L∞(Ω), s∈(1,2) and f∈C(Ω×R,R). the exact assumptions on a and f are as follows:(a) there exists a positive measure subset K(?)Ω, a(x)>0in K.(f1) uniformly in x∈Ω.(f2) uniformly in x∈Ω.The main results of the present paper are the following:Theorem1Suppose (a) is satisfied, assume f(x,t) satisfies (f1) and the following condition:(f3) uniformly in x∈Ω. Then problem (P) has a nontrivial solution.Theorem2Suppose (a) is satisfied, assume f(x,t) satisfies (f1) and the following conditions:(f4) uniformly in x∈Ω;(f5) uniformly in x∈Ω. Then problem (P) has a nontrivial solution. Theorem3Suppose (a) are satisfied, assume f(x,t) satisfies (f2) and thefollowing condition:(f6) uniformly in x∈Ω and b≠λi\(λi-c). Then there exists a constant m0>0such that problem (P) has two nontrivial solutions for‖a‖∞≤m0.Theorem4Suppose (a) are satisfied, assume f(x,t) satisfies (f2) and the following conditions:(f7) uniformly in x∈Ω;Us) uniformly in x∈Ω. Then there exists a constant m0>0such that problem (P) has two nontrivial solutions for‖a‖∞≤m0.Theorem5Suppose (a) are satisfied, assume f(x,t) satisfies (f2) and the following conditions:(f9) uniformly in x∈Ω;(f10) uniformly in x∈Ω, for some constant r>0, r∈(1,N+4/N-4);(f11) uniformly in x∈Ω, where max{N/4(r-1),s}<σ<2N/N-4. Then there exists a constant m0>0such that problem (P) has two nontrivial solutions for‖a‖∞≤m0.Theorem6Suppose a(x)=0, f(x,t) is odd function in t, assume that f(x, t) satisfies (f9) and the following condition:(f12) there exists C>0such that|f(x,t)|≤C(1+|t|τ), where r∈(1,N+4/N-4).(f13) there exist a≥1and b≥0such that H(x, t)≤aH(x, s)+b, where H(x,ti)=f(x, t)t-2F(x, t) and s≤t≤0or0≤t≤s. Then problem (P) has infinity many high energy solutions.