Positive solutions of some semilinear elliptic equations in R+n with Neumann boundary conditions

In this paper, we consider positive classical solutions of {−△u=f(u),x=(x′,t)∈R+n,n≥3∂u∂t=h(u),on ∂R+n where f(s)f(s) is locally bounded in (0,∞)(0,∞) and f(s)s−(1+4ν) is non-increasing in (0,∞)(0,∞) for some ν>n−2ν>n−2, h(s)h(s) is locally bounded in (0,∞)(0,∞) and h(s)s−(1+2ν) is non-decreasing in (0,∞)(0,∞) for the same νν. We get that the possible solution only depends on tt, and several corollaries that include previous results of various authors are established.