Existence of solutions for Kirchhoff type problems with critical nonlinearity in \({\mathbb{R}^N}\)

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Abstract

In this paper, we consider the existence and multiplicity of standing wave solutions of Kirchhoff type problems with critical nonlinearity in \({\mathbb{R}^N}\) :
$$-\varepsilon^p \left(a + b \int\limits_{\mathbb{R}^N} \frac{1}{p}|\nabla u|^p{\rm d}x \right) \,{\rm div}(|\nabla u|^{p-2}\nabla u) + V(x)|u|^{p-2}u = K(x)|u|^{p^\ast-2}u + h(x,u),$$
for all \({(t, x) \in \mathbb{R} \times \mathbb{R}^N}\), where V(x) is a nonnegative potential, and K(x) is a bounded positive function. Under suitable assumptions, we show that this equation has at least one solution provided that \({\varepsilon < \mathcal {E}}\), for any \({m \in \mathbb{N}}\), it has m pairs of solutions if \({\varepsilon < \mathcal {E}_m}\), where \({\mathcal {E}}\) and \({\mathcal {E}_m}\) are sufficiently small positive numbers. Moreover, these solutions \({u_\varepsilon \rightarrow 0}\) in \({W^{1,p}(\mathbb{R}^N)}\) as \({\varepsilon \rightarrow 0}\).

Mathematics Subject Classification (2000)

58E05 58E50 

Keywords

Kirchhoff type problems Critical nonlinearity Variational method Semiclassical states 
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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.College of MathematicsChangchun Normal UniversityChangchunPeople’s Republic of China
  2. 2.Institute of Mathematics, School of Mathematical ScienceNanjing Normal UniversityNanjingPeople’s Republic of China
  3. 3.Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of EducationJilin UniversityChangchunPeople’s Republic of China

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