Energy Estimates and Existence Results for a Mixed Boundary Value Problem for a Complete Sturm-Liouville Equation Exploiting a Local Minimization Principle

In our work, we are going to look for local minima for the Euler functional corresponding to a mixed boundary value problem for a complete Sturm-Liouville equation where the coefficients can also be negative, to obtain the existence results and energy estimates for solutions for the problem. In particular, we establish the existence of a non-zero solution for a specific localization of the parameter and show that the solution exists for positive values of the parameter, under the condition that the nonlinear component exhibits sublinearity both at the origin and at infinity. The proof relies on a local minimum theorem for differentiable functionals. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz.