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

Abstract

We prove an Ambrosetti–Prodi type result for the third order fully nonlinear equation u ‴ ( t ) + f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) = s p ( t ) with f : [ 0 , 1 ] × R 3 → R and p : [ 0 , 1 ] → R + continuous functions, s ∈ R , 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.