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 $-\left(|\phi_x|^{p-2}\phi_x\right)_x=\lambda \left(|\phi|^{q-2}\phi-f(\phi)\right)$ 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 $1<p\neq 2$, the existing results treat only the case when $f$ is either odd and a power (see \cite{TAYA}) or when $p=q$ (\cite{Guedda-Veron}). Our method of proof rely on a careful analysis of the phase diagram associated with this equation, refining the regularity results of \cite{otani} and characterizing the exact points where we may have $C^2$ regularity of solutions including some points $\chi\in (0,1)$ for which $\phi_x(\chi)=0$.