Entire positive solutions for an inhomogeneous semilinear biharmonic equation

Abstract

In this paper, we investigate the entire positive solutions for the inhomogeneous biharmonic equation (∗) − Δ 2 u + u p + f ( x ) = 0 in R n , where Δ 2 is the biharmonic operator, p > 1 , n ≥ 5 and 0 ⁄ ≡ f ∈ C ( R n ) 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 f for existence and nonexistence. And also, we obtain that there exist at least two types of decay solutions at infinity with the assumptions on f .