Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian and application

2016/09/18 by A. Aghajani, Aghajani, Asadollah, Alireza Tehrani +1
Computer Science · Mathematics · #35B32 #35B40 #35J91 #35P30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1609.05437

openalex publication_date 2016/09/18 · openalex created_date 2016/09/30 · openalex updated_date 2026/07/28

Abstract

We derive a priori bounds for positive supersolutions of - Δ_p u = ρ(x) f(u) , where p>1 and Δ_p is the p-Laplace operator, in a smooth bounded domain of R^N with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem - Δ_p u = λf(u) , with Dirichlet boundary condition, where f is a nondecreasing continuous differentiable function on [0,∞] such that f(0) > 0 , f(t)^(1)/(p-1) is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter λ_p^* . In particular, we consider the nonlinearities f(u) = e^u and f(u) = (1+u)^m ( m > p-1 ) and give explicit estimates on λ_p^* . As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p -Laplacian that improves obtained results in the recent literature for some range of p and N .

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