<p>This paper is devoted to the study of a specific class of one-dimensional stationary free boundary problems characterized by variable coefficients. Such problems arise frequently in various physical and engineering applications, where the location of the boundary is an unknown that must be determined as part of the solution. Our primary objective is to address the fundamental questions of existence and uniqueness in a rigorous mathematical framework. First, we establish the existence of solutions by employing a penalty-based regularization technique in conjunction with the Schauder fixed point theorem. This approach allows us to handle the nonlinearities inherent in free boundary formulations. Furthermore, a detailed analysis is conducted to investigate the regularity of the solutions and the topological properties of the resulting free boundary. Finally, by introducing a specific monotonicity assumption on the variable coefficients, we provide a formal proof for the uniqueness of the solution. The results obtained in this work extend certain classical existence theories to more general cases involving non-constant coefficients, providing deeper insights into the behavior of stationary free boundary models.</p>