Variational approaches for sixth-order boundary value problems

In this work, we obtain the existence of multiple solutions for sixth-order differential equations with two parameters λ and μ, generally arising in astrophysics; the narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modeled by the problems. Using a consequence of the local minimum theorem due to Bonanno we look into the existence of one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due to Averna and Bonanno, and another one due to Bonanno we guarantee the existence of two and three solutions for the problem in a special case.