Multiple solutions for a class of non-local problems for semilinear elliptic equations
type: Item
instance of: scholarly article
title: Multiple solutions for a class of non-local problems for semilinear elliptic equations
zbMATH Open document ID: 0823.35055
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1994-04-12T00:00:00Z
review text: We investigate the solvability of the following non-local problem for the semilinear elliptic equation \[ \begin{aligned} Lu=- \sum_{i,j=1}^ n a_{ij} (x) D_{ij} u+ \sum_{i=1}^ n b_ i(x) D_ i u+ c(x)u= g(x,u)+ t\Phi (x)+ f(x) &\quad \text{in } Q,\\ u(x)- \beta(x) u(\varphi (x)) =0 &\quad \text{on } \partial Q, \end{aligned} \tag \(N_ t\) \] in a bounded domain \(Q\) with a smooth boundary \(\partial Q\), where \(\varphi: \partial Q\to Q\) and \(\beta: \partial Q\to \mathbb{R}\) are given functions and \(t\) is real parameter. The problem of this type is often referred to as the boundary value problem with the Bitsadze-Samarskii condition. The most characteristic feature of a non-local problem is that the boundary condition relates values of a solution on the boundary to its values on some parts of the interior of the region. The main purpose of this article is to prove the existence result for the problem \((N_ t)\) under the assumption of the Ambrosetti-Prodi type.
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 548918
zbMATH Keywords: non-local problem
zbMATH Keywords: semilinear elliptic equation
zbMATH Keywords: Bitsadze-Samarskii condition
zbMATH Keywords: assumption of the Ambrosetti-Prodi type
author: Jan Chabrowski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195168852
OpenAlex ID: W2024473016
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DOI: 10.2977/PRIMS/1195168852
description: scientific article; zbMATH DE number 548918