A bifurcation problem for a one-dimensional p-Laplace elliptic problem with non-odd absorption

Abstract

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem −(|ϕx|p−2ϕx)x=λ(|ϕ|q−2ϕ−f(ϕ)) such that ϕ(0)=ϕ(1)=0, where p,q>1, λ is a positive real parameter and f is a continuous (not necessarily odd) function. Our goal is to give a complete description of solutions of this problem. We completely characterize the set of solutions of this problem, which may be uncountable. For 1<p≠2, the existing results treat only the case when f is either odd and a power (see [11]) or when p=q ([8]). Our method of proof relies on a careful analysis of the phase diagram associated with this equation, refining the regularity results of M. Ôtani in 1984 (see [10]) and characterizing the exact points where we may have C2 regularity of solutions including some points χ∈(0,1) for which ϕx(χ)=0.