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

Authors :
de Araujo, Anderson L. A.
Ercole, Grey
Vargas, Julio C. Lanazca
Publication Year :
2023

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.<br />Comment: 19 pages

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2303.00140
Document Type :
Working Paper