The Existence Of Nontrivial Solutions To Biharmonic Equation Without (AR) Condition On Bounded Domain

In this paper, we study the existence of nontrivial solution for the following biharmonic problem:where Ω is a unit ball in RN(N≥ 5). We assume that the nonlinearity f(x, u) satisfies the following conditions:(Hi) (ⅰ) f: x R → R is a Caratheodory function;(ⅰⅱ) f(x,s)s≥ 0, (?)(x, 5) €Ω×R and f(x,0)≡0,(?)x∈Ω.(H2) There is a p∈ (2,2N/N-4), such that(H3) lim f(x,s)/s=0, uniformly in x∈Ω(H4) There is a∈(0,∞), such thatBy using the monotone method developed by Jeanjean in [7], we prove the exis-tence of nontrivial solution of the above biharmonic equation under (H1)-(H4) for different parameter a.