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arXiv · 1811.11350

Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent

Yuan Li·Dun Zhao·Qingxuan Wang
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Abstract

We study the following nonlinear Hartree-type equation \begin{equation*} -Δu+V(x)u-a(\frac{1}{|x|^γ}\ast |u|^2)u=λu,~\text{in}~\mathbb{R}^N, \end{equation*} where $a>0$, $N\geq3$, $γ\in(0,2)$ and $V(x)$ is an external potential. We first study the asymptotic behavior of the ground state of equation for $V(x)\equiv1$, $a=1$ and $λ=0$ as $γ\nearrow2$. 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 $γ\nearrow2$, which leads to symmetry breaking. Moreover, the concentration rate for maximum points of ground states will be given.

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Yuan Li, Dun Zhao, Qingxuan Wang. 2018-11-28. Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent. https://doi.org/10.1007/s00033-019-1172-5

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