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

In this paper, we study the following problemequation(0.1){−Δu+u=k(x)f(u)+h(x),x∈RN,u∈H1(RN),u>0in RN,N⩾3, where f(t)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)00 and some θ∈(0,12), as well as the monotonicity of f(t)/tf(t)/t are not assumed. Under appropriate assumptions on k,hk,h and f, we prove that problem (0.1) has at least two positive solutions.