Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhoff term
type: Item
instance of: scholarly article
title: Positive solutions of logistic equations with dependence on gradient and nonhomogeneous Kirchhoff term
zbMATH Open document ID: 1351.35063
author: Nguyen Bich Huy
author: Bui The Quan
published in: Journal of Mathematical Analysis and Applications
publication date: 2016-07-29T00:00:00Z
review text: The paper is concerned with the existence of positive solutions for the following type elliptic equation: \[ -M(x,\|u\|)\Delta_p u = \lambda f(x, u,\nabla u) - g(x, u,\nabla u) \text{ in } \Omega,\;u|_{\partial \Omega} = 0, \;u \in W^{1,p}_0 (\Omega), \leqno{(P)} \] where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, \(\Delta_p u = \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) with \(p\in (1,N)\) , \(\|\cdot \|\) is the norm of Sobolev space \(W^{1,p}_0 (\Omega)\) , \(\lambda\) is a real parameter, \(M \in C( \Omega \times \mathbb{R}^+ , \mathbb{R}^+)\), and \(f, g : \Omega \times \mathbb{R}^+ \times \mathbb{R}^N \rightarrow \mathbb{R}^+\) are Caratheodory functions. By reducing the equation (P) to a fixed point problem, and based on the fixed point index theory and the cone theoretic arguments, the authors prove that: (i) In the ``non-degenerate'' case of the function \(M\), the equation (P) has a positive solution if either \(\lambda > 0\) and \(f(x, s,t)\) is \((p - 1)\)-sublinear in \(s\), or \(\lambda >\lambda_1\) and \(f(x, s,t)\) is \((p -1)\)-linear in \(s\). (ii) In the case of ``degenerate'' of the function \(M\), the equation (P) has a positive solution for \(\lambda \in (\lambda_1, \lambda_2)\) and has no positive solution if \(\lambda > \lambda_2 + \sigma\) if \(f(x, s,t)\) is \((p - 1)\)-linear in \(s\), for some \(\sigma\). Here, \(\lambda_1, \lambda_2\) are the principal eigenvalues of some \(p-\)Laplacian problems with zero Dirichlet boundary data.
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 6607764
zbMATH Keywords: positive solution
zbMATH Keywords: logistic equation
zbMATH Keywords: non-homogeneous Kirchhoff term
zbMATH Keywords: fixed point index
zbMATH Keywords: cone theoretic argument
reviewed by: Huansong Zhou
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.06.020
OpenAlex ID: W2435460452
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DOI: 10.1016/J.JMAA.2016.06.020
description: scientific article; zbMATH DE number 6607764