Infinite semipositone problems with asymptotically linear growth forcing terms.
type: Item
instance of: scholarly article
title: Infinite semipositone problems with asymptotically linear growth forcing terms.
zbMATH Open document ID: 1274.35089
author: D. D. Hai
author: Lakshmi Sankar
author: R. Shivaji
published in: Differential and Integral Equations
publication date: 2013-03-21T00:00:00Z
review text: Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N\), let \(p\in (1,\infty )\), \(\alpha \in (0,1)\) and \(\lambda >0\), and let \(f:[0,\infty [\rightarrow \mathbb {R}\) be a continuous function asymptotically \(p\)-linear at \(\infty \). Consider the following boundary value problem involving a singular elliptic equation \[ \Delta _p u=\lambda f(u)-u^{-\alpha }\;\text{ in } \Omega ,\; \; u_{\mid \partial \Omega }=0, \] where \(\Delta _p\) is the \(p\)-Laplacian operator. Using a sub-supersolutions method, the authors prove that there exists a compact interval \(I\subset (0,\infty )\) such that for every \(\lambda \in I\), the above problem admits at least one positive solution. The same result is also established in the case in which \(\Omega \) is the exterior of a ball, and the nonlinearity \(\lambda f(u)-u^{-\alpha }\) is multiplied by a continuous radially symmetric function \(K\: \Omega \rightarrow \mathbb {R}\) which vanishes at \(\infty \). In this case, the authors prove the existence of a positive radially symmetric solution. Finally, with some additional conditions on the nonlinearities, the above results are extended to systems.
reviewed by: Giovanni Anello
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J75
Mathematics Subject Classification ID: 35J92
zbMATH DE Number: 6148072
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: singular elliptic problem
zbMATH Keywords: sub-supersolutions method
zbMATH Keywords: positive solution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6148072