Boundary value problems for some fourth order ordinary differential equations

Abstract

We consider beam equations of the form

with boundary conditions corresponding either to a simply-supported or a clamped beam. The function α is supposed to be nonnegative and we assume that the underlying linear problem has zero as its first eigenvalue. We give a unified treatment of the existence of solutions for both boundary conditions, using variational or degree-theoretic methods.

Keywords:

• Resonance
• variational properties of eigenvalues
• variational methods

AMS (MOS)::

• 34B15
• 34B25
• 58E30