Multiple standing waves of Helmholtz equation with mixed dispersion concentrating in the high frequency limit

In this paper, we study the nonlinear Helmholtz equation with mixed dispersion \begin{equation*} Δ^2 u-βk^2, Δu+αk^4 u=W(x), |u|^{p-2}u~$\text{in}$~$\mathbb{R}$^N, \end{equation*} where the weight function $W(x)$ is continuous, nonnegative, and satisfies $$\underset{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}W(x)<\underset{x\in {\mathbb{R}}^N}{\sup }W(x)\mathrm{.}$$ Within each of the following parameter ranges, \begin{center} (a) $\alpha <0$, $\beta \in \mathbb{R}$; \qquad (b) $\alpha >0$, $\beta <-2\sqrt{\alpha }$; \qquad © $\alpha =0$, $\beta <0$, \end{center} After a suitable rescaling, we obtain the existence of dual ground state solutions, which concentrate along the global maximizers of $W$ as $k\to \mathrm{\infty }$. In addition, we establish the existence of multiple solutions associated with the set of global maximum points of $W$, and we further characterize the precise concentration behavior of these solutions.