Positive Entire Solutions For An Inhomogeneous Semilinear Elliptic Equation

In this paper, we consider the existence, nonexistence and decay properties of positive entire solutions for the inhomogeneous semilinear elliptic equationwhere n > 3, p > 1, f(x) > 0 as well as K(|x|) is a given locally Holder continuous function in Rn\{0}, and -K(|z|) satisfies the slow decay condition, i.e., K(|x|) > C|x|l near oo for some constant C > 0 and l > -2.The main difficulties in studying existence, nonexistence and other related properties of positive entire solutions for (*) are the noncompactness of the domain, the presence of an inhomogeneous term and the possible appearance of singular coefficients, which greatly restrict the use of regular methods. The existence and nonexistence of positive entire solutions have been firstly obtained under some different assumptions on f(x). Then we prove that the decay exponent of any positive entire solution for equation (*) lies between (2+l)/(p-1) and n - 2. and fast decay solutions (i.e. the decay exponent is n - 2) are existent. We finally prove the existence of solutions lying in the neighborhood of ua.The results of this paper show that if 1 < p < (n+1)/(n-2), then equation (*) does not possess any positive entire solution. If p > (n+1)/(n-2^, the decay exponent of f(x) is very sharp for the positive entire solutions of equation (*): if the decay exponent of f(x) is smaller than (2p+l)/(p-1), then equation (*) does not have any positive entire solution; however, if the decay exponent of f(x) is not smaller than (2p+l)/(p-1), then equation (*) does have at least one positive entire solution; furthermore, if p and the decay exponent of f(x) is bigger than n, then the decay exponent of any positive entire solution must be one of the two extremes, i.e., or n - 2.