Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}

Liang, Sihua; Shi, Shaoyun

doi:10.1063/1.4850835

Title: Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}

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Abstract

Using variational methods, we establish existence of multi-bump solutions for a class of Kirchhoff type problems −(a+b∫{sub R{sup 3}}|∇u|{sup 2}dx)Δu+λV(x)u=f(u), where f is a continuous function with subcritical growth, V(x) is a critical frequency in the sense that inf{sub x∈R{sup 3}}V(x)=0. We show that if the zero set of V(x) has several isolated connected components Ω{sub 1}, …, Ω{sub k} such that the interior of Ω{sub i} is not empty and ∂Ω{sub i} is smooth, then for λ > 0 large there exists, for any non-empty subset J ⊂ (1, …, k), a bump solution is trapped in a neighborhood of ∪{sub j∈J}Ω{sub j}.

Authors:

Liang, Sihua ^[1]; Shi, Shaoyun ^[2]

College of Mathematics, Changchun Normal University, Changchun 130032, Jilin (China)
College of Mathematics, Jilin University, Changchun 130012 (China)

Publication Date:: Sun Dec 15 00:00:00 EST 2013

OSTI Identifier:: 22250912

Resource Type:: Journal Article

Journal Name:: Journal of Mathematical Physics

Additional Journal Information:: Journal Volume: 54; Journal Issue: 12; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICAL METHODS AND COMPUTING; CRITICAL FREQUENCY; FUNCTIONS; MATHEMATICAL SOLUTIONS; TRAPPING; VARIATIONAL METHODS

Citation Formats


                    Liang, Sihua, College of Mathematics, Jilin University, Changchun 130012, Shi, Shaoyun, and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012. Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}.  United States: N. p., 2013. 
        Web.  doi:10.1063/1.4850835.

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                    Liang, Sihua, College of Mathematics, Jilin University, Changchun 130012, Shi, Shaoyun, & Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012. Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}.  United States.  https://doi.org/10.1063/1.4850835

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                    Liang, Sihua, College of Mathematics, Jilin University, Changchun 130012, Shi, Shaoyun, and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012. 2013.  
        "Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}".  United States.  https://doi.org/10.1063/1.4850835.

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@article{osti_22250912,

  title        = {Existence of multi-bump solutions for a class of Kirchhoff type problems in R{sup 3}},

  author       = {Liang, Sihua and College of Mathematics, Jilin University, Changchun 130012 and Shi, Shaoyun and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012},

  abstractNote = {Using variational methods, we establish existence of multi-bump solutions for a class of Kirchhoff type problems −(a+b∫{sub R{sup 3}}|∇u|{sup 2}dx)Δu+λV(x)u=f(u), where f is a continuous function with subcritical growth, V(x) is a critical frequency in the sense that inf{sub x∈R{sup 3}}V(x)=0. We show that if the zero set of V(x) has several isolated connected components Ω{sub 1}, …, Ω{sub k} such that the interior of Ω{sub i} is not empty and ∂Ω{sub i} is smooth, then for λ > 0 large there exists, for any non-empty subset J ⊂ (1, …, k), a bump solution is trapped in a neighborhood of ∪{sub j∈J}Ω{sub j}.},

  doi          = {10.1063/1.4850835},

  url          = {https://www.osti.gov/biblio/22250912},
  journal      = {Journal of Mathematical Physics},

  issn         = {0022-2488},

  number       = 12,

  volume       = 54,

  place        = {United States},

  year         = {Sun Dec 15 00:00:00 EST 2013},

  month        = {Sun Dec 15 00:00:00 EST 2013}

}

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