Existence and Multiplicity Results for a Mixed Sturm-liouville Type Boundary Value Problem

where the potentials are given functions. Under various boundary conditions, Sturm and Liouville established that solutions of problem (1) can exist only for particular values of the real parameter λ, which is called an eigenvalue. Relevant examples of linear Sturm-Liouville problems are the Bessel equation and the Legendre equation. The classical Sturm-Liouville theory does not depend upon the calculus of variations, but stems from the theory of ordinary linear or nonlinear differential equations. Linear Sturm-Liouville equations can be also studied in the context of functional analysis by means of self-adjoint operators or integral operators with a continuous symmetric kernel (the Green’s function of the problem). Certain applications involving linear partial differential equations can be treated with the help of the Sturm-Liouville theory, for instance the normal modes of vibration of a thin membrane. We also refer to [20], where it is studied a perturbed nonlinear SturmLiouville problem with superlinear convex nonlinearity. In the recent paper [16], the authors study a class of discrete anisotropic Sturm-Liouville problems. In the present paper, we are concerned with a class of nonlinear SturmLiouville problems and we establish some qualitative properties of the eigenvalues