On some third order nonlinear boundary value problems: existence, location and multiplicity results

Abstract

We prove an Ambrosetti–Prodi type result for some third order fully nonlinear equations, with a parameter s,under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u'' is obtained. An existence and location result will be proved, by degree theory, for s ∈ R such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies.