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 ? is a unit ball in RN(N? 5). We assume that the nonlinearity f(x, u) satisfies the following conditions:(Hi) (?) f: x R ? R is a Caratheodory function;(??) f(x,s)s? 0, (?)(x, 5) €?×R and f(x,0)?0,(?)x??.(H2) There is a p? (2,2N/N-4), such that(H3) lim f(x,s)/s=0, uniformly in x??(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.