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

## TL;DR

This paper investigates the behavior of solutions to a nonlinear p-Laplacian problem with convection and exponential terms, proving existence, convergence to boundary distance, and analyzing the effects of parameters on solutions.

## Contribution

It establishes the existence of positive solutions under certain parameters and demonstrates uniform convergence of solutions to the boundary distance function as p approaches infinity.

## Key findings

- Existence of positive solutions for certain parameter ranges.
- Solutions converge uniformly to the boundary distance function as p increases.
- New convergence result for problems involving nonlinearities with convection terms.

## Abstract

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.