The existence and non-existence of solutions for the nonlinear Schrödinger equation in one dimension

We study the following nonlinear Schrödinger equations in one dimension: (∗)−u′′+(1+b(x))u=f(u),u∈H1(R),where b(x) satisfies 1+b(x)≥0, lim|x|→∞b(x)=0 and lim¯|x|→∞eβ0|x|b(x)≤0 for β0>2 and f(u) satisfies some properties whose typical example is f(u)=|u|p−1u (p>1). In this paper, we give conditions of b(x) where (∗) has a positive solution and a condition on b(x) where (∗) has no nontrivial solutions.