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Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent

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Abstract

We study the following nonlinear Hartree-type equation

$$\begin{aligned} -\Delta u+V(x)u-a\left( \frac{1}{|x|^\gamma }*|u|^2\right) u=\lambda u,~\text {in}~{\mathbb {R}}^N, \end{aligned}$$

where \(a>0\), \(N\ge 3\), \(\gamma \in (1,2)\) and V(x) is an external potential. We first study the asymptotic behavior of the ground state of equation for \(V(x)\equiv 1\), \(a=1\) and \(\lambda =0\) as \(\gamma \nearrow 2\). Then, we consider the case of some trapping potential V(x) and show that all the mass of ground states concentrate at a global minimum point of V(x) as \(\gamma \nearrow 2\), which leads to symmetry breaking. Moreover, the concentration rate for maximum points of ground states will be given.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Yuan Li & Dun Zhao

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, People’s Republic of China

    Qingxuan Wang

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  1. Yuan Li
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  2. Dun Zhao
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  3. Qingxuan Wang
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Correspondence to Qingxuan Wang.

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Qingxuan Wang: This work is supported by the NSFC Grants 11801519 and 11475073.

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Li, Y., Zhao, D. & Wang, Q. Concentration behavior of nonlinear Hartree-type equation with almost mass critical exponent. Z. Angew. Math. Phys. 70, 128 (2019). https://doi.org/10.1007/s00033-019-1172-5

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  • DOI: https://doi.org/10.1007/s00033-019-1172-5

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