We study the behavior, as \(p\rightarrow \infty ,\) of solutions to the Dirichlet problem \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u=u^{q(p)-1}+\mu _{p}u^{a(p)-1}\left| \nabla u\right| ^{r(p)-a(p)} & \quad \text {in}\ \Omega \\ u>0 & \quad \text {in}\ \Omega \\ u=0 & \quad \text {on} \ \partial \Omega . \end{array} \right. \end{aligned}\) We assume that q(p), a(p) and r(p) are continuous functions of p satisfying \(1\le q(p)<p\text { and }1\le a(p)\le r(p)\) for every p sufficiently large, and such that \(0<Q<1\text { and }0\le A\le R<\infty ,\) where \(\begin{aligned} Q:=\lim _{p\rightarrow \infty }\frac{q(p)}{p},~~A:=\lim _{p\rightarrow \infty }\frac{a(p)}{p}\text { and }R:=\lim _{p\rightarrow \infty }\frac{r(p)}{p}. \end{aligned}\) As for the parameter \(\mu _{p}\) we assume that \(\begin{aligned} 0<\Lambda :=\lim _{p\rightarrow \infty }(\mu _{p})^{1/p}<\Lambda _{\infty } ^{A+\frac{Q(R-1)}{1-Q}} \end{aligned}\) where \(\Lambda _{\infty }:=\left\| d_{\Omega }\right\| _{\infty }^{-1}\) and \(d_{\Omega }\) denotes the distance function to the boundary of \(\Omega .\)
