Existence of Sign-Changing Solutions to Equations Involving the One-Dimensional p-Laplacian

Abstract

We consider the equations involving the one-dimensional p-Laplacian (P): (u′tp-2u′(t))′+λf(u(t))=0, 0<t<1, and u(0)=u(1)=0, where p>1,λ>0,f∈C1(R;R),f(s)s>0, and s≠0. We show the existence of sign-changing solutions under the assumptions f∞=lim|s|→∞⁡(fs/sp-1)=+∞ and f0=lim|s|→0(f(s)/sp-1)∈[0,∞]. We also show that (P) has exactly one solution having specified nodal properties for λ∈(0,λ*) for some λ*∈(0,∞). Our main results are based on quadrature method.