Biharmonic Equation Superlinear Multi-solution Results

In this paper, we give some multiplicity results on existence of a biharmonic elliptic problemwith superlinear terms f(x,u).△² denotes the biharmonic operator andΩ, is an open bounded domain in R^N with smooth boundary (?)Ω,. c <λ₁, whereλ₁ is the first eigenvalue of the eigenvalue problemf(x, u) satisfies conditions as follow :(f₁)f∈C¹(Ω×R,R).(f₂)f(x,0)=0=f_u(x,0).(f₃) |f(x,u)|≤C(1 + |u|^p-1) ,(?)x∈Ω,u∈R, where C > 0, 2 < p<2N/N-4, if N≥4; p > 2, if N < 4.(f₄) There isμ> 2, M > 0, such that, for |u|≥M,(f₅) F(x, u)≥0, for all x, u. Moreover uf(x, u) > 0 for |u| > 0 small enough.(f'₅) uf(x, u)≥2F(x, u)≥0, for all x and u, and the first inequality is strict for |u| > 0 small enough.(f₆) F(x, u)≤, 0, for |u| > 0 small enough.Let 0 <λ₁ <λ₂ <…<λ_k <…be a sequence of distinguished eigenvalues of (0.2). Denote by The main results in this paper are:Theorem A Assume f satisfy (f₁)-(f₅) and k≥1 be fixed. Then there existsδ> 0 such that forλ∈(Λ_k+1-δ,Λ_k+1), problem (0.1) has at least two nontrivial solutions, if (f₅) is replaced by (f'₅), we will get the third solution.Theorem B Assume (f₁)- (f₄) and (f₆), let k≥1 be fixed. Then there existsδ> 0 such that for sup_{(x,u)∈Ω×R} F^-(x,u) <δandλ∈(Λ_k+1-δ,Λ_k+1], problem (0.1) has at least two nontrivial solutions.