Variational method for the existence of weak solutions of boundary value problems of nonlinear elliptic partial differential equations

In this study, we investigate the application of variational methods to establish the existence of weak solutions for boundary value problems (BVPs) associated with nonlinear elliptic partial differential equations (PDEs). By employing the direct method in the calculus of variations, we reformulate the given PDEs as minimisation problems over suitable Sobolev spaces, where weak solutions correspond to critical points of appropriately defined energy functionals. The analytical framework incorporates foundational tools such as Sobolev embeddings, compactness theorems, and properties of coercivity and weak lower semi-continuity to ensure the well-posedness of the variational formulation. We present and prove an existence theorem under well-defined conditions on the nonlinear coefficients of the equations, specifically those involving the Carathéodory property, polynomial growth bounds, monotonicity, and symmetry. These conditions are shown to satisfy the hypotheses of the Fucik-Kufner theorem, which guarantees the existence - and, under convexity assumptions, uniqueness - of weak solutions. To demonstrate the utility of the theoretical results, we apply the theorem to selectednonlinear elliptic problems and verify the fulfilment of all required criteria for the existence of solutions. Through this approach, we provide a rigorous and general framework for analysing nonlinear elliptic BVPs without resorting to classical linearization techniques. The findings confirm that variational methods offer powerful, flexible tools for addressing existence questions in complex PDE systems, with relevance to mathematical modelling in physics, engineering, and applied sciences.