Bifurcation of positive entire solutions for a semilinear elliptic equation

In this paper, we consider the nonhomogeneous semilinear elliptic equation (*)λ -△u + u = λK(x)u^p + h(x) in ℝ ^N,u ε H¹ in (ℝⁿ) where λ ≥ 0, 1 < p < (N + 2)/(N - 2), if N ≥ 3, 1 < p < ∞, if N = 2, h(x) ε H^-1(ℝ^N), 0 ≢ h(x) ≥ 0 in ℝ^N, K(x) is a positive, bounded and continuous function on ℝ^N. We prove that if K(x) ≥ K_∞ > 0 in ℝ^N, and lim K(x) = K∞, |x|-∞ then there exists a positive constant λ* such that (*)λ has at least two solutions if λ π (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*. Copyright Clearance Centre, Inc.