Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator

Zedong Yang; Guotao Wang; Ravi P. Agarwal; Haiyong Xu

doi:10.3934/dcdss.2020436

Article Contents

2021, Volume 14, Issue 10: 3821-3836. Doi: 10.3934/dcdss.2020436

This issue Previous Article Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function Next Article On the observability of conformable linear time-invariant control systems

Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator

a.
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China
b.
Department of Mathematics, Texas A & M University, Kingsville, TX 78363-8202, USA
c.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
d.
College of Science & Technology, Ningbo University, Ningbo, Zhejiang 315211, China

^* Corresponding author: Ravi P. Agarwal
^* Corresponding author: Ravi P. Agarwal

All authors equally contributed this manuscript

Received: March 31, 2020

Revised: May 31, 2020

Early access: November 2020

Published: October 2021

This work is supported by NSFC (No.11501342), NSF of Shanxi, China (No.201701D221007), Science and Technology Innovation Project of Shanxi Normal University (No.2019XSY027), the Graduate Innovation Program of Shanxi, China (No.2020SY337) and STIP (Nos.201802068 and 201802069)

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Abstract

In this paper, we study the positive solutions of the Schrödinger elliptic system

$ \begin{equation*} \begin{split} \left\{\begin{array}{ll}{\operatorname{div}(\mathcal{G}(|\nabla y|^{p-2})\nabla y) = b_{1}(|x|) \psi(y)+h_{1}(|x|) \varphi(z),}& {x \in \mathbb{R}^{n}(n \geq 3)}, \\ {\operatorname{div}(\mathcal{G}(|\nabla z|^{p-2})\nabla z) = b_{2}(|x|) \psi(z)+h_{2}(|x|) \varphi(y),} & {x \in \mathbb{R}^{n}},\end{array}\right. \end{split} \end{equation*} $

where $ \mathcal{G} $ is a nonlinear operator. By using the monotone iterative technique and Arzela-Ascoli theorem, we prove that the system has the positive entire bounded radial solutions. Then, we establish the results for the existence and nonexistence of the positive entire blow-up radial solutions for the nonlinear Schrödinger elliptic system involving a nonlinear operator. Finally, three examples are given to illustrate our results.

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Mathematics Subject Classification: Primary: 35A24, 35B09; Secondary: 35B44.

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