Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities |
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Authors: | Giovany M Figueiredo Norihisa Ikoma João R Santos Júnior |
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Institution: | 1. Faculdade de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém, Pará, Brazil 2. Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai-shi, Miyagi, 980-8578, Japan 3. Faculdade de Matemática, Universidade Federal do Pará, 66.075-110, Belém, Pará, Brazil
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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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