Clustered helical vortices for 3D incompressible Euler equation in infinite cylinders

In this article, we first consider solutions to a semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2$\text{div}$(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\ u=0,\ \ &x\in\partial Ω\end{cases} \end{equation*} for small values of $\epsilon$. We prove that there exists a family of clustered solutions which have arbitrary many bubbles and collapse into given maximum points of $q^2\sqrt{\det K}$ as $\epsilon \to 0.$ Then as an application, we construct clustered traveling-rotating helical vortex solutions to Euler equations in infinite cylinders, such that the support set of corresponding vortices consists of several helical tubes concentrating near a single helix.