Solvability of boundary value problem for Schrödinger Equations with Reverse Hölder Potentials on $L^p$ and endpoint spaces

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schrödinger-type equation $-\text{DIV}(A(x)\mathrm{\nabla }u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$ independent of $t$ and $V$ in Reverse Hölder class ${\mathcal{B}}_q$, and Neumann boundary data ${\mathrm{\partial }}_{{\nu }_A}u(x,0)=f(x)\in H_{\mathcal{L}}^p(\text{rn})$, or Regularity data $u(x,0)=g\in H_V^{1,p}(\text{rn})$, utilizing the method of layer potential. We prove the solvability when $A$ is a small $L^{\mathrm{\infty }}$ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds. Besides we also give the Campanato norm estimate of the double layer potential related to the Dirichlet problem with boundary data in certain Campanato-type spaces.