Nonexistence of positive solutions to n-th order equations in Rⁿ

In this paper, we are mainly concerned with the following integral equations: [Formula presented] where n≥2, γ∈R, u∈C(Rⁿ) and f(x,u) may change signs and satisfies some assumptions. By using the method of scaling spheres developed by Dai and Qin in [14], we first derive nonexistence of positive solutions to the above IEs under some assumptions. Then, based on the equivalence between the above IEs and the following 2D PDEs: −Δu(x)=f(x,u),x∈R², we also obtain nonexistence of positive solutions to the 2D PDEs under some assumptions. One should note that there are no growth conditions on u and hence f(x,u) can grow exponentially (or even faster) on u.