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

In this paper we study the existence of solutions of a one-dimensional eigenvalue problem -(|& phi;x|p-2 & phi;x)x = & lambda; |& phi;|q-2 & phi; - f (& phi;) such that & phi;(0) = & phi;(1) = 0, where p, q > 1, & lambda; 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. Otani in 1984 (see [10]) and characterizing the exact points where we may have C2 regularity of solutions including some points & chi; & ISIN; (0, 1) for which & phi;x (& chi; ) = 0. & COPY; 2023 Elsevier Inc. All rights reserved.