ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

Huyuan Chen; Feng Zhou

doi:10.3934/DCDSS.2023035

ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

Huyuan Chen
, Feng Zhou^*

^*Corresponding author for this work

School of Mathematical Sciences

Jiangxi Normal University

Research output: Contribution to journal › Article › peer-review

Abstract

Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation [Formula presented] where N ≥ 2, p > 1 and the potential [Formula presented] is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin’s subcritical case and then isolated singular solutions are derived with the form u_k = kΦ + v_k via the Schauder fixed point theorem for the integral equation [Formula presented] in [Formula presented], where Φ is the real valued fundamental solution −∆ − 1 and wσ is also a real valued solution of (−∆ − 1)wσ = δ0 with the asymptotic behavior at infinity controlled by [Formula presented] for some [Formula presented].

Original language	English
Pages (from-to)	3035-3056
Number of pages	22
Journal	Discrete and Continuous Dynamical Systems - Series S
Volume	16
Issue number	11
DOIs	https://doi.org/10.3934/DCDSS.2023035
State	Published - 2023

Keywords

Helmholtz equation, isolated singularity

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title = "ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION",

abstract = "Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation [Formula presented] where N ≥ 2, p > 1 and the potential [Formula presented] is a H{\"o}lder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin{\textquoteright}s subcritical case and then isolated singular solutions are derived with the form uk = kΦ + vk via the Schauder fixed point theorem for the integral equation [Formula presented] in [Formula presented], where Φ is the real valued fundamental solution −∆ − 1 and wσ is also a real valued solution of (−∆ − 1)wσ = δ0 with the asymptotic behavior at infinity controlled by [Formula presented] for some [Formula presented].",

keywords = "Helmholtz equation, isolated singularity",

author = "Huyuan Chen and Feng Zhou",

year = "2023",

doi = "10.3934/DCDSS.2023035",

language = "英语",

volume = "16",

pages = "3035--3056",

journal = "Discrete and Continuous Dynamical Systems - Series S",

issn = "1937-1632",

publisher = "American Institute of Mathematical Sciences",

number = "11",

}

TY - JOUR

T1 - ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

AU - Chen, Huyuan

AU - Zhou, Feng

PY - 2023

Y1 - 2023

N2 - Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation [Formula presented] where N ≥ 2, p > 1 and the potential [Formula presented] is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin’s subcritical case and then isolated singular solutions are derived with the form uk = kΦ + vk via the Schauder fixed point theorem for the integral equation [Formula presented] in [Formula presented], where Φ is the real valued fundamental solution −∆ − 1 and wσ is also a real valued solution of (−∆ − 1)wσ = δ0 with the asymptotic behavior at infinity controlled by [Formula presented] for some [Formula presented].

AB - Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation [Formula presented] where N ≥ 2, p > 1 and the potential [Formula presented] is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin’s subcritical case and then isolated singular solutions are derived with the form uk = kΦ + vk via the Schauder fixed point theorem for the integral equation [Formula presented] in [Formula presented], where Φ is the real valued fundamental solution −∆ − 1 and wσ is also a real valued solution of (−∆ − 1)wσ = δ0 with the asymptotic behavior at infinity controlled by [Formula presented] for some [Formula presented].

KW - Helmholtz equation, isolated singularity

UR - https://www.scopus.com/pages/publications/85179164562

U2 - 10.3934/DCDSS.2023035

DO - 10.3934/DCDSS.2023035

M3 - 文章

AN - SCOPUS:85179164562

SN - 1937-1632

VL - 16

SP - 3035

EP - 3056

JO - Discrete and Continuous Dynamical Systems - Series S

JF - Discrete and Continuous Dynamical Systems - Series S

IS - 11

ER -

ON ISOLATED SINGULAR SOLUTIONS OF SEMILINEAR HELMHOLTZ EQUATION

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