On the existence of positive solutions to a certain class of semilinear elliptic equations

Norihisa Ikoma

doi:10.1007/s42985-021-00079-7

On the existence of positive solutions to a certain class of semilinear elliptic equations

Norihisa Ikoma

Department of Mathematics

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

2 Citations (Scopus)

Abstract

In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

Original language	English
Article number	28
Journal	Partial Differential Equations and Applications
Volume	2
Issue number	2
DOIs	https://doi.org/10.1007/s42985-021-00079-7
Publication status	Published - 2021 Apr

Keywords

Concentration-compactness lemma
Mountain pass theorem
Positive solution

ASJC Scopus subject areas

Analysis
Applied Mathematics
Computational Mathematics
Numerical Analysis

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10.1007/s42985-021-00079-7

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@article{c82d106c0f4a41afb2b695721cbd7a18,

title = "On the existence of positive solutions to a certain class of semilinear elliptic equations",

abstract = "In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.",

keywords = "Concentration-compactness lemma, Mountain pass theorem, Positive solution",

author = "Norihisa Ikoma",

note = "Funding Information: The author would like to thank Lawrence Craig Evans for pointing out the variational structure of () to him. He also would like to thank Hitoshi Ishii for helpful comments on the consideration of (). The author was supported by JSPS KAKENHI Grant Numbers JP 17H02851, 19H01797 and 19K03590. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.",

year = "2021",

month = apr,

doi = "10.1007/s42985-021-00079-7",

language = "English",

volume = "2",

journal = "Partial Differential Equations and Applications",

issn = "2662-2963",

publisher = "Springer International Publishing AG",

number = "2",

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T1 - On the existence of positive solutions to a certain class of semilinear elliptic equations

AU - Ikoma, Norihisa

N1 - Funding Information: The author would like to thank Lawrence Craig Evans for pointing out the variational structure of () to him. He also would like to thank Hitoshi Ishii for helpful comments on the consideration of (). The author was supported by JSPS KAKENHI Grant Numbers JP 17H02851, 19H01797 and 19K03590. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.

PY - 2021/4

Y1 - 2021/4

N2 - In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

AB - In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

KW - Concentration-compactness lemma

KW - Mountain pass theorem

KW - Positive solution

UR - https://www.scopus.com/pages/publications/85126327021

UR - https://www.scopus.com/pages/publications/85126327021#tab=citedBy

U2 - 10.1007/s42985-021-00079-7

DO - 10.1007/s42985-021-00079-7

M3 - Article

AN - SCOPUS:85126327021

SN - 2662-2963

VL - 2

JO - Partial Differential Equations and Applications

JF - Partial Differential Equations and Applications

IS - 2

M1 - 28

ER -

On the existence of positive solutions to a certain class of semilinear elliptic equations

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Keywords

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