Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian
Date
2017-02-14
Authors
Aghajani, Asadollah
Mosleh Tehrani, Alireza
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We derive a priori bounds for positive supersolutions of -∆pu = ρ(x)ƒ(u), where p > 1 and ∆p is the p-Laplace operator, in a smooth bounded domain of ℝN with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem -∆pu = λƒ(u), with Dirichlet boundary condition, where ƒ is a nondecreasing continuous differentiable function on such that ƒ(0) > 0, ƒ(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 ƒ(u) = eu and ƒ(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.
Description
Keywords
Nonlinear eigenvalue problem, Estimates of principal eigenvalue, Extremal parameter
Citation
Aghajani, A., & Mosleh Tehrani, A. (2017). Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian. <i>Electronic Journal of Differential Equations, 2017</i>(46), pp. 1-14.
Rights
Attribution 4.0 International
Rights Holder
This work is licensed under a Creative Commons Attribution 4.0 International License.