Four positive solutions for the semilinear elliptic equation: - Δu + u = a(x)u^p + f(x) in ℝ^N

We consider the existence of positive solutions of the following semilinear elliptic problem in ℝ^N: (Formula Presented) where 1 < p < N + 2/N - 2 (N ≥ 3), 1 < p < ∞ (N = 1, 2), a(x) ∈ C(ℝ^N), f(x) ∈ H^-1 (ℝ^N) and f(x) ≥ 0. Under the conditions: 1° a(x) ∈ (0, 1) for all x ∈ ℝ^N, 2° a(x) → 1 as |x| → ∞, 3° there exist δ > 0 and C > 0 such that a(x) - 1 ≥ -Ce^-(2+δ)|x| for all x ∈ ℝ^N, 4° a(x) ≢ 1, we show that (*) has at least four positive solutions for sufficiently small ||f||_H-1(ℝN) but f ≢ 0.