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Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type

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Abstract

In this paper, we deal with the following singularly perturbed elliptic problem

$$\begin{array}{ll}-\varepsilon^2\Delta u+V(x)u=f(u),\quad u \in H^1(\mathbb{R}^2),\end{array}$$

where f(s) has critical growth of Trudinger–Moser type. In this paper, we construct a localized bound-state solution concentrating at an isolated component of the positive local minimum points of V as \({\varepsilon \rightarrow 0}\) under certain conditions on f(s). Our results complete the analysis made in Byeon et al. (Commun Partial Differ Equ 33: 1113–1136, 2008) for the two-dimensional case, in the sense that, in that paper only the subcritical growth was considered.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, People’s Republic of China

    Jianjun Zhang

  2. Chern Institute of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China

    Jianjun Zhang

  3. Department of Mathematics, Federal University of Paraíba, 58051-900, João Pessoa, PB, Brazil

    João Marcos do Ó

Authors
  1. Jianjun Zhang
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  2. João Marcos do Ó
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Correspondence to João Marcos do Ó.

Additional information

Research partially supported by The National Institute of Science and Technology of Mathematics ICNT-Mat, CAPES and CNPq/Brazil. J. Zhang was partially supported by CAPES/Brazil and CPSF (2013M530868).

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Zhang, J., do Ó, J.M. Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger–Moser type. Z. Angew. Math. Phys. 66, 3049–3060 (2015). https://doi.org/10.1007/s00033-015-0565-3

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  • DOI: https://doi.org/10.1007/s00033-015-0565-3

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