Positive solutions for nonlinear elliptic problems of [formula omitted]-Laplacian type on [formula omitted] without (AR) condition

Abstract

In this paper, we study the following problem (P){−Δpu+|u|p−2u=α(x)f(u),x∈RN,u∈W1,p(RN),u(x)>0 in RN, where 1<p≤N and α is a positive bounded function. For the case when f(t) is asymptotically linear at infinity, the Ambrosetti–Rabinowitz type condition and the monotonicity of f(t)t are not assumed; for the case of superlinear at infinity, the Ambrosetti–Rabinowitz type condition is not assumed. Under appropriate assumptions on α and f, we prove that problem (P) has at least one positive solution.