Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Existence of a positive solution of an elliptic equation on RN

Zhaoli Liua1 and Zhi-Qiang Wanga2

a1 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

a2 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou 35007, People's Republic of China

Abstract

We prove the existence of a positive solution for the Schrödinger-type equation on RN,−Δu + V(x)u = f(u) for u xs2208 H1(RN), provided that

S0308210500003152disp001

The result is obtained in two cases. (a) lim|x|→ ∞ V(x) = V (∞) xs2208 (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*.

(Received July 10 2003)

(Accepted October 07 2003)