The aim of the monograph is to present new existence and the multiplicity results in the study of sublinear elliptic problems in different contexts, combining techniques from calculus of variations, geometry, and from the theory of symmetrizations, based on works by the author. In the first part of the work, we introduce the basic definitions and results from the theory of Lebesgue spaces and Sobolev spaces, from the theory of calculus of variations, from the theory of locally Lipschitz functions, and finally from Riemann/Finsler geometry. In the second part, we present some new existence and multiplicity results in the Euclidean setting. In the third part of the present monograph, we present some nonlinear problems on Riemannian manifolds. It is well known that variational methods are indispensable in geometry. In the last part, we consider nonlinear PDEs on Finsler manifolds, providing existence, uniqueness/multiplicity results.