The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

Abstract

We consider, for $a,l\ge 1,$ $b,s,\alpha >0,$ and $p>q\ge 1,$ the homogeneous Dirichlet problem for the equation $-{\Delta }_{p}u=\lambda u^{q-1}+\beta u^{a-1}{\left|\nabla u\right|}^b+m{u}^{l-1}{e}^{\alpha u^s}$ 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 $p\to \infty .$ This convergence result is new for nonlinearities involving a convection term.

19 pages