Abstract
In this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.
Highlights
In this article, we consider the following Schrödinger boundary value problems of second order: ifs + f = –|f |pf, (t, s) ∈ Rn × [0, L), (1.1)f (0, t) = f0(t), √where i = –1, n ∂2 = i=1 ∂ti2 is the Laplace operator in Rn, f (t, s) : Rn × [0, L) → C denotes the complex valued function, L is the maximum existence time, n is the space dimension and p satisfies the embedding condition ⎧ ⎨+∞, n < p ⎩ 4 n–2, n = 1, 2, n > 2. (2020) 2020:1
A lot of attention is paid to the existence and nonexistence of global solutions to the second-order boundary value problems related to the Schrödinger equation
In 2019, Xue and Tang [49] established the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in Rn
Summary
1 Introduction In this article, we consider the following Schrödinger boundary value problems of second order (see [1,2,3,4, 12, 24, 29, 33, 38]): ifs + f = –|f |pf , (t, s) ∈ Rn × [0, L), (1.1) A lot of attention is paid to the existence and nonexistence of global solutions to the second-order boundary value problems related to the Schrödinger equation. Boundary value problems driven by a combination of differential operators of different nature (such as (p, 2)-equations) were studied in [30].
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