Understanding pattern formation in nature is an important topic in applied mathematics. For more than three decades, the Ohta–Kawasaki energy has attracted considerable attention from applied mathematicians. This energy functional, which combines surface energy and electrostatic potential energy, captures the intricate patterns observed in various physical and biological systems. Despite its apparent simplicity, the Ohta–Kawasaki energy serves as a versatile framework for describing a wide range of pattern formation phenomena induced by competing interactions. In this dissertation, we aim to gain a better understanding of the important properties of the Ohta–Kawasaki energy, specifically its stationary points, global minimizers, and energy landscape. We explore these properties in the context of broad applications such as nuclear physics, block copolymers, and biological membranes. In order to investigate the complicated geometries in these applications, we utilize asymptotic analysis and numerical simulations. Firstly, we explore the stationary points of the Ohta–Kawasaki energy. Specifically, we study how a three-dimensional ball loses stability as the nonlocal coefficient increases in the binary case. Our approach combines numerical simulations and bifurcation analysis. We calculate the minimum energy path for the transition from a single ball to two separate balls, as well as the bifurcation branch orginating from the ball. In the context of nuclear physics, this bifurcation branch is known as the Bohr–Wheeler branch. Our simulations suggest that, unlike the previous understanding, all the stationary points on this bifurcation branch are unstable. Similar results are observed in two dimensions. This finding illustrates the unexpected mechanism governing the stability loss of balls and disks. Secondly, we explore the global minimizers of the Ohta–Kawasaki energy. We numerically compute the one-dimensional energy minimizers of relatively short patterns in the non-degenerate ternary case. Inspired by our numerical results, we propose an array of periodic candidates. We then show that our candidates can have lower energy than the previously conjectured global minimizer which is of the cyclic pattern. Our results are consistent with simulations based on other theories and physical experiments of triblock copolymers, in which noncyclic lamellar patterns have been found. This finding indicates that even in one dimension, the global minimizers of the Ohta–Kawasaki energy can exhibit unexpected richness. Lastly, we explore the energy landscape of the Ohta–Kawasaki energy. We propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model in the degenerate ternary case. Our phase-field simulations and asymptotic results suggest that the limit of the recovery sequence exhibits behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes. This finding reveals the intricate landscape of the Ohta–Kawasaki energy. In summary, this dissertation sheds light on three important aspects of the Ohta–Kawasaki energy: its stationary points, global minimizers, and energy landscape. Our findings are timely contributions to the ongoing research on pattern formation driven by energetic competition.