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Multiplicity and concentration behavior of solutions to the critical Kirchhoff-type problem

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Abstract

In this paper, we study the multiplicity and concentration phenomenon of positive solutions to the critical Kirchhoff-type problem:

$$\begin{aligned} -\left( \varepsilon ^2 a+\varepsilon b\int \limits _{\mathbb {R}^3}|\nabla u|^2{{\mathrm {d}}} x\right) \Delta u + V(x) u = f(u)+h(x) u^5 \quad {\mathrm{in }} \quad \mathbb {R}^3, \end{aligned}$$

where \(\varepsilon \) is a small positive parameter, a, b are positive constants, \(V \in C({\mathbb {R}}^3,\mathbb {R})\) is a positive potential, \(f \in C^1(\mathbb {R}^+, \mathbb {R})\) is a subcritical nonlinear term, \(h(x)u^5\) is a critical nonlinearity. When \(\varepsilon >0\) small, we establish the relationship between the number of positive solutions and the profile of V and h. The concentration behavior and some further properties of positive solutions are also obtained.

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Authors and Affiliations

  1. College of Science, China University of Petroleum, Qingdao, 266580, Shandong, People’s Republic of China

    Jian Zhang

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Wenming Zou

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  1. Jian Zhang
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  2. Wenming Zou
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Correspondence to Jian Zhang.

Additional information

Supported by NSFC (11401583), the Fundamental Research Funds for the central Universities (16CX02051A) and Shandong Provincial Natural Sciences Foundation (ZR2015AL006).

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Zhang, J., Zou, W. Multiplicity and concentration behavior of solutions to the critical Kirchhoff-type problem. Z. Angew. Math. Phys. 68, 57 (2017). https://doi.org/10.1007/s00033-017-0803-y

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  • DOI: https://doi.org/10.1007/s00033-017-0803-y

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