Positive entire stable solutions of inhomogeneous semilinear elliptic equations

For n≥3n≥3 and p>1p>1, the elliptic equation Δu+K(x)up+μf(x)=0Δu+K(x)up+μf(x)=0 in Rn possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions KK and ff vanish rapidly, for instance, K(x),f(x)=O(|x|l)K(x),f(x)=O(|x|l) near ∞∞ for some l<−2l<−2 and (ii) μ≥0μ≥0 is sufficiently small. Especially, in the radial case with K(x)=k(|x|)K(x)=k(|x|) and f(x)=g(|x|)f(x)=g(|x|) for some appropriate functions k,gk,g on [0,∞)[0,∞), there exist two intervals Iμ,1Iμ,1, Iμ,2Iμ,2 such that for each α∈Iμ,1α∈Iμ,1 the equation has a positive entire solution uαuα with uα(0)=αuα(0)=α which converges to l∈Iμ,2l∈Iμ,2 at ∞∞, and uα1