On Positive Weak Solutions for Some Nonlinear Elliptic Boundary Value Problems Involving the p-Laplacian

Volume 3, Issue 1, pp 94--101 http://dx.doi.org/10.22436/jmcs.03.01.08

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Authors

S. H. Rasouli - Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran G. A. Afrouzi - Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran J. Vahidi - Department of Applied Mathematics, Iran University of Science and Technology, Behshahr, Iran

Abstract

This study concerns the existence of positive weak solutions to boundary value problems of the form \[ \begin{cases} -\Delta_{p}u=g(x,u),\,\,\,\,\, x\in \Omega,\\ u(x)=0,\,\,\,\,\, x\in \partial \Omega, \end{cases} \] where \(\Delta_{p}\) is the so-called p-Laplacian operator i.e. \(\Delta_pz=div(|\nabla z|^{p-2} \nabla z), p>1,\Omega\) is a smooth bounded domain in \(R^N(N\geq 2)\) with \(\partial \Omega\) of class \(C^2\); and connected, and \(g(x; 0) < 0\) for some \(x\in \Omega\) (semipositone problems). By using the method of sub-super solutions we prove the existence of the positive weak solution to special types of \(g(x; u)\).

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Keywords

• Positive weak solutions
• p-Laplacian
• sub-super solution

MSC

•  35D30
•  35J92
•  35J57

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