Bifurcation of positive entire solutions for a semilinear elliptic equation

In this paper, we consider the nonhomogeneous semilinear elliptic equation,where λ ≥ 0, 1 < p < (N + 2)/(N − 2), if N ≥ 3, 1 < p < ∞, if N = 2, h(x) ∈ H−l(ℝN), 0 ≢ h(x) ≥ 0 in ℝN, K(x) is a positive, bounded and continuous function on ℝN. We prove that if K(x) ≥ K∞ > 0 in ℝN, and lim∣x∣⃗∞K(x) = K∞, then there exists a positive constant λ✶ such that (✶)λ has at least two solutions if λ ∈ (0, λ✶) and no solution if λ > λ✶. Furthermore, (✶)λ has a unique solution for λ = λ✶ provided that h(x) satisfies some suitable conditions. We also obtain some further properties and bifurcation results of the solutions of (1.1)λ at λ = λ✶.