Entire positive solutions for an inhomogeneous semilinear biharmonic equation

In this paper, we investigate the entire positive solutions for the inhomogeneous biharmonic equation equation(∗)−Δ2u+up+f(x)=0in Rn, where Δ2Δ2 is the biharmonic operator, p>1p>1, n≥5n≥5 and 0⁄≡f∈C(Rn)0⁄≡f∈C(Rn) is a given nonnegative function. Based on the results on the biharmonic equation in [Q.Y. Dai, Entire positive solutions for inhomogeneous semilinear elliptic systems, Glasgow Math. J 47 (2005) 97–114], we obtain the optimal “decay coefficient” of the inhomogeneous term ff for existence and nonexistence. And also, we obtain that there exist at least two types of decay solutions at infinity with the assumptions on ff.