Electron. J. Differential Equations, Vol. 2017 (2017), No. 46, pp. 1-14.

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

Asadollah Aghajani, Alireza Mosleh Tehrani

Abstract:
We derive a priori bounds for positive supersolutions of $-\Delta_p u = \rho(x) f(u)$, where p>1 and $\Delta_p$ is the p-Laplace operator, in a smooth bounded domain of $\mathbb{R}^N$ with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem $-\Delta_p u = \lambda f(u)$, with Dirichlet boundary condition, where $f$ is a nondecreasing continuous differentiable function on 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 $\lambda_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 $\lambda_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.

Submitted June 10, 2015. Published February 14, 2017.
Math Subject Classifications: 35J66, 35J92, 35P15.
Key Words: Nonlinear eigenvalue problem; estimates of principal eigenvalue; extremal parameter.

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Asadollah Aghajani
School of Mathematics
Iran University of Science and Technology
Narmak, Tehran 16844-13114, Iran.
phone +9821-73913426. Fax +9821-77240472
email: aghajani@iust.ac.ir
Alireza Mosleh Tehrani
School of Mathematics
Iran University of Science and Technology
Narmak, Tehran 16844-13114, Iran
email: amtehrani@iust.ac.ir

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