  We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous
Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta
u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a
smooth bounded domain $\Omega\subset\mathbb{R}^{N}.$ We prove that under
certain setting of the parameters $\lambda,$ $\beta$ and $m$ the problem admits
at least one positive solution. Using this result we prove that if
$\lambda,\beta>0$ are arbitrarily fixed and $m$ is sufficiently small, then the
problem has a positive solution $u_{p},$ for all $p$ sufficiently large. In
addition, we show that $u_{p}$ converges uniformly to the distance function to
the boundary of $\Omega,$ as $p\rightarrow\infty.$ This convergence result is
new for nonlinearities involving a convection term.
