The existence and non-existence of solutions for the nonlinear Schrödinger equation in one dimension. (October 2018) - Sato, Yohei
Abstract: We study the following nonlinear Schrödinger equations in one dimension: ( ∗ ) − u ′ ′ + ( 1 + b ( x ) ) u = f ( u ), u ∈ H 1 ( R ), where b ( x ) satisfies 1 + b ( x ) ≥ 0, lim | x | → ∞ b ( x ) = 0 and lim ¯ | x | → ∞ e β 0 | x | b ( x ) ≤ 0 for β 0 > 2 and f ( u ) satisfies some properties whose typical example is f ( u ) = | u | 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.