Existence of a positive solution of an elliptic equation on RN

Abstract

We prove the existence of a positive solution for the Schrödinger-type equation on RN,−Δu + V(x)u = f(u) for u ∈ H1(RN), provided that The result is obtained in two cases. (a) lim|x|→ ∞V(x) = V (∞) ∈ (0,∞), f is subcritical and supt>0 2F(t)t−2 < V (∞). (b) There exist V > 0 and R* > 0 such that f(t)t−1 < V ∞ ≤ V (x) for all t > 0 and |x| ≥ R*.