Multiple solutions for second-order Sturm-Liouville boundary value problems with subquadratic potentials at zero (Q2052076)

Summary: We deal with the following Sturm-Liouville boundary value problem: \[ \begin{cases} -(P(t) x^\prime(t))^\prime + B (t)x(t) = \lambda \nabla_x V(t, x),\quad\text{a.e. } t \in [0,1], \\ x(0) \cos\alpha - P(0) x^\prime(0) \sin\alpha = 0, \\ x (1) \cos\beta - P(1) x^\prime(1) \sin\beta = 0. \end{cases} \] Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm-Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.