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

We study the following nonlinear Schrodinger equations in one dimension: -u '' + (1+b(x))u = f(u), u is an element of H-1 (R) where b(x) satisfies 1+ b(x) >= 0, lim vertical bar x vertical bar ->infinity b(x) = 0 and (lim) over bar vertical bar x vertical bar ->infinity e(beta 0 vertical bar x vertical bar)b(x) <= 0 for < 0 for beta(0) > 2 and f (u) satisfies some properties whose typical example is f(u) = vertical bar u vertical bar(p-1)u (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. (C) 2018 Elsevier Ltd. All rights reserved.