We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Elliptic PDEs with the Critical Sobolev Exponent. To this end, we first recall some useful results from functional analysis, including the Sobolev spaces, which provide a natural setting for the idea of weak or generalised solutions. We then present linear PDE theory, including eigenvalues of the Dirichlet Laplacian operator. We discuss the Direct Methods of Calculus of Variations and Critical Point Theory, together with examples of how these techniques are applied to solving PDEs. We show how the existence of solutions to semilinear elliptic equations depends on the exponent of the growth of the non-linear term. This then naturally leads to the discussion of the critical Sobolev exponent, where we present both positive and negative results
