A fourth order BVP of Sturm-Liouville with asymmetric unbounded nonlinearities

Abstract

It is obtained an existence and location result for the fourth order boundary value problem of Sturm-Liouville type u^{(iv)}(t)=f(t,u(t),u′(t),u′′(t),u′′′(t)), for t∈[0,1], u(0)=u(1)=A, k₁u′′′(0)-k₂u′′(0)=0, k₃u′′′(1)+k₄u′′(1)=0, where f:[0,1]×R⁴→R is a continuous function and A,k_{i}∈R, for i=1,...,4, are such that k₁,k₃>0, k₂,k₄≥0. We assume that f verifies a one-sided Nagumo type growth condition which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments make use of an a priori estimate on the third derivative of a class of solutions, the lower and upper solutions method and degree theory.