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Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities

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Abstract

In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations

$$- \varepsilon^2 M \left(\varepsilon^{2-N}\!\!\int_{\mathbf{R}^N}|\nabla u|^2\,\mathrm{d} x \right) \Delta u \!+\! V(x) u \!=\! f(u) \quad{\rm in}\ \mathbf{R}^N, \quad u \!\in\! H^1(\mathbf{R}^N), \ N \!\geqq\!1,$$

where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions \({(u_\varepsilon)_{\varepsilon \in (0,\tilde{\varepsilon}]}}\) which concentrates at a local minimum of V after extracting a subsequence (ε k ).

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Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém, Pará, Brazil

    Giovany M. Figueiredo

  2. Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai-shi, Miyagi, 980-8578, Japan

    Norihisa Ikoma

  3. Faculdade de Matemática, Universidade Federal do Pará, 66.075-110, Belém, Pará, Brazil

    João R. Santos Júnior

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  1. Giovany M. Figueiredo
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  2. Norihisa Ikoma
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  3. João R. Santos Júnior
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Correspondence to Norihisa Ikoma.

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Communicated by P. Rabinowitz

Giovany M. Figueiredo: Supported by PROCAD/CASADINHO: 552101/2011-7, CNPq/PQ 301242/2011-9 and CNPQ/CSF 200237/2012-8.

Norihisa Ikoma: Partially supported by JSPS Research Fellowships 24-2259.

João R. Santos Júnior: Partially supported by CAPES - Brazil - 7155123/2012-9.

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Figueiredo, G.M., Ikoma, N. & Santos Júnior, J.R. Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities. Arch Rational Mech Anal 213, 931–979 (2014). https://doi.org/10.1007/s00205-014-0747-8

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