MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION∗

doi:10.1007/s10473-023-0421-z

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1819-1840.doi: 10.1007/s10473-023-0421-z

MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

Yansheng ZHONG^†, Yongqing LI

School of Mathematics and Statistics & FJKLMAA, & Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou 350117, China

收稿日期:2022-02-12 修回日期:2022-10-19 发布日期:2023-08-08
通讯作者: †Yansheng ZHONG, E-mail: zys08@fjnu.edu.cn
作者简介:Yongqing LI, E-mail: yqli@fjnu.edu.cn
基金资助:
*Partially supported by the NSFC (11671085) and the Science Foundation of Fujian Province (2020J01160).

MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

Yansheng ZHONG^†, Yongqing LI

School of Mathematics and Statistics & FJKLMAA, & Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou 350117, China

Received:2022-02-12 Revised:2022-10-19 Published:2023-08-08
Contact: †Yansheng ZHONG, E-mail: zys08@fjnu.edu.cn
About author:Yongqing LI, E-mail: yqli@fjnu.edu.cn
Supported by:
*Partially supported by the NSFC (11671085) and the Science Foundation of Fujian Province (2020J01160).

摘要/Abstract

摘要： The existence and multiplicity of positive solutions for equation (1.1) with the new critical exponent $4<p<2\cdot 2^*$ shall be investigated in a high dimension. The conclusions extend the relative results recently attained in \cite{1} for the one-dimensional case. More precisely, as the coefficient $a(x)$ in the nonlinearity is sign-changing, the modified term $2(\Delta (|u|^2))u$ is still helpful for obtaining multiple positive solutions in a high dimension, even if a sign condition like $\int_{\mathbb{R}^N}a(x)e_1^p{\rm d}x<0$ (also named “a necessary condition” see [2,3]) does not hold.

关键词: variational methods, nonlinear Schrö, dinger equation, sign-changing

Abstract: The existence and multiplicity of positive solutions for equation (1.1) with the new critical exponent $4<p<2\cdot 2^*$ shall be investigated in a high dimension. The conclusions extend the relative results recently attained in \cite{1} for the one-dimensional case. More precisely, as the coefficient $a(x)$ in the nonlinearity is sign-changing, the modified term $2(\Delta (|u|^2))u$ is still helpful for obtaining multiple positive solutions in a high dimension, even if a sign condition like $\int_{\mathbb{R}^N}a(x)e_1^p{\rm d}x<0$ (also named “a necessary condition” see [2,3]) does not hold.

Key words: variational methods, nonlinear Schrö, dinger equation, sign-changing

Yansheng ZHONG, Yongqing LI. MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1819-1840.

Yansheng ZHONG, Yongqing LI. MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1819-1840.

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MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHRÖDINGER EQUATION IN A HIGH DIMENSION^∗

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