On concentration of real solutions for fractional Helmholtz equation

Zifei Shen; Shuijin Zhang

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arXiv · 2312.15009

On concentration of real solutions for fractional Helmholtz equation

Zifei Shen·Shuijin Zhang

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Abstract

This paper studies the nonlinear fractional Helmholtz equation \begin{equation}\label{main} (-\Delta)^{s} u-k^{2} u=Q(x)|u|^{p-2}u, ~~\mathrm{in}~~\mathbb{R}^{N},~~N\geq3, \end{equation} where $\frac{N}{N+1} 0$ large, the existence of real-valued solutions for (\ref{main}) are proved, and in the limit $k\longrightarrow\infty$, sequence of solutions associated with ground states of a dual equation are shown to concentrate, after rescaling, at global maximum points of the function $Q$.

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Zifei Shen, Shuijin Zhang. 2023-12-20. On concentration of real solutions for fractional Helmholtz equation. https://arxiv.org/abs/2312.15009

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