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

2023/03/01 by Anderson L. A. de Araújo, de Araujo, Anderson L. A., Grey Ercole +3
Computer Science · Mathematics · #35B40 #35J92 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2303.00140

openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider, for a,l≥1, b,s,α>0, and p>q≥1, the homogeneous Dirichlet problem for the equation -Δ_pu=λu^q-1+βu^a-1\vert ∇ u\vert ^b+mu^l-1e^αu^s in a smooth bounded domain Ω⊂ℝ^N. We prove that under certain setting of the parameters λ, β and m the problem admits at least one positive solution. Using this result we prove that if λ,β>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 Ω, as p→∞. This convergence result is new for nonlinearities involving a convection term.

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