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

Abstract

We investigate the existence and nonexistence of solutions to the Dirichlet problem \begin{equation*} \tag{$P$} \label{pba} \left\{ \begin{alignedat}{2} -Δ_p u + g(u) |\nabla u|^p &= λf(u) \quad &&\mbox{in} \;\; Ω, \\ u &= 0 \quad &&\mbox{on} \;\; \partialΩ, \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 $f(0)\ge 0$, 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 $u\ge 0$ and $\Vert u{\Vert }_{C(\overline{\Omega })}\in (\alpha ,\beta ]$, 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 $(\alpha ,\beta ]$.