Nonexistence of nontrivial solutions to Kirchhoff-like equations
type: Item
instance of: scholarly article
title: Nonexistence of nontrivial solutions to Kirchhoff-like equations
author: Christopher Goodrich
published in: Proceedings of the American Mathematical Society. Series B
publication date: 2024-07-16T00:00:00Z
review text: Denoting by \(*\) the usual convolution on \([0,1]\), the author consider the one-dimensional parameterized Kirchhoff equation \N\[\N-M \left ( (a * |u|^q )(1)\right ) u^{ \prime \prime}(t)= \lambda f(t,u(t)), \quad 1<t<1. \N\]\NHere, the coefficient function \(M\) is continuous, \(a \in L^1(0,1)\) is positive and \(q>0\). Making use of the Green's function \(G:[0,1] \times [0,1] \to [0, \infty)\) associated to the operator \(-u^{\prime \prime}\) subject to given boundary data, the main result of the paper provides sufficient conditions for the existence of some \(\lambda_0>0\) such that the above equation associated with the boundary data implied by \(G\), has no nontrivial positive solution for \(\lambda > \lambda_0\).
reviewed by: P. Jebelean
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 42A85
Mathematics Subject Classification ID: 44A35
zbMATH DE Number: 7881500
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: convolution
zbMATH Keywords: Green's function
zbMATH Keywords: Harnack constant
MaRDI profile type: Publication
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DOI: 10.1090/BPROC/224
description: scientific article; zbMATH DE number 7881500