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.
Categories: math.AP