ON THE FOURTH ORDER SEMIPOSITONE PROBLEM IN RN

For N≥ 5 and α > 0, we consider the following semipositone Problem ( equation presented) where g ϵ L1 loc(RN) is an indefinite weight function, fα : R → R is a continuous function that satisfies fα(t) = -α for t ϵ R-, and D2,2(RN) is the completion of C∞ c (RN) with respect to ( ∫ RN (Δu)2)1/2. For fα satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of α1 > 0 such that for each α ϵ (0, α1), (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if α is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in (0, α1), relying on the Riesz potential of the biharmonic operator.