Positive solutions for a nonhomogeneous elliptic equation on without (AR) condition

In this paper, we study the following problem(0.1) { − Δ u + u = k ( x ) f ( u ) + h ( x ) , x ∈ R N , u ∈ H 1 ( R N ) , u > 0 in R N , N ⩾ 3 , where f ( t ) is either asymptotically linear or superlinear with respect to t at infinity. The Ambrosetti–Rabinowitz type condition, that is so-called (AR) condition: (AR) 0 < F ( t ) ≜ ∫ 0 t f ( s ) d s ⩽ θ t f ( t ) , for t > 0 and some θ ∈ ( 0 , 1 2 ) , as well as the monotonicity of f ( t ) / t are not assumed. Under appropriate assumptions on k , h and f, we prove that problem (0.1) has at least two positive solutions.