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Topological Methods in Nonlinear Analysis

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

Authors

  • Anderson L. A. de Araujo https://orcid.org/0000-0002-3640-9794
  • Grey Ercole https://orcid.org/0000-0002-0459-7292
  • Julio C. Lanazca Vargas

DOI:

https://doi.org/10.12775/TMNA.2023.061

Keywords

Convection term, distance function, exponential term, gradient estimate

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.

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Published

2024-09-21

How to Cite

1.
DE ARAUJO, Anderson L. A., ERCOLE, Grey and VARGAS, Julio C. Lanazca. The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms. Topological Methods in Nonlinear Analysis. Online. 21 September 2024. Vol. 64, no. 1, pp. 339 - 359. [Accessed 6 July 2025]. DOI 10.12775/TMNA.2023.061.
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Vol 64, No 1 (September 2024)

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Copyright (c) 2024 Anderson L. A. de Araujo, Grey Ercole, Julio C. Lanazca Vargas

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This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.

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