Multiple sign-changing solutions to the Sturm–Liouville boundary value problem with resonance

Abstract

In this paper, we study the Sturm–Liouville boundary value problem -(p(x)u′)′+q(x)u=f(x,u),x∈[0,1] subject to αu(0)-βu′(0)=γu(1)+δu′(1)=0. By constructing a new Sobolev space Hα,γ1[0,1], we discuss the existence of multiple solutions, especially the existence of multiple sign-changing solutions to this problem when the nonlinear f is resonant both at 0 and ∞. By combining the methods of the Morse theory, the topological degree and the fixed point index, we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions. If this problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions.