Regularizing effect of the natural growth term in quasilinear problems with sign-changing nonlinearities

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -\Delta_p u + g(u) |\nabla u|^p &= \lambda f(u) \quad &&\mbox{in} \;\; \Omega, \\ u &= 0 \quad &&\mbox{on} \;\; \partial\Omega, \end{alignedat} \right. \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a smooth bounded domain, $p\in (1,\infty)$, $\lambda>0$ and $g\in C(\mathbb{R})$. Our main assumption is that $:f \mathbb{R}\to \mathbb{R}$ is a continuous function such that $f(s)>0$ for all $s\in (\alpha,\beta)$, where $0<\alpha<\beta$ are two zeros of $f$. If $f(0)\geq 0$, we show that an area condition involving $f$ and $g$ is both sufficient and necessary in order to have a pair $(\lambda,u)\in \mathbb{R}^+\times C_0^1(\overline{\Omega})$, with $u\geq 0$ and $\|u\|_{C(\overline{\Omega})}\in (\alpha,\beta]$, solving~\eqref{pba}. We also study how the presence of the gradient term affects the existence of solution. Roughly speaking, the more negative $g$ is, the stronger its regularizing effect on~\eqref{pba}. We prove that, regardless of the shape of $f$, for any fixed $\lambda$, there always exists a function $g$ such that~\eqref{pba} admits a nonnegative solution with maximum in $(\alpha,\beta]$.