Positive stationary solutions of convection-diffusion equations for superlinear sources

IF 1 Q1 MATHEMATICS
A. Orpel
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引用次数: 1

Abstract

We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation \[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\] for \(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when \(f(x,\cdot)\) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient \(g\) without radial symmetry.
超线性源对流扩散方程的正平稳解
研究了一类外域对流扩散方程正平稳解的存在性和多重性。这个问题导致如下的椭圆方程\[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\]对于\(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\)。本文的目的是证明我们的问题在无穷邻域中具有有限能量的极小解的不可减数列。我们还证明了这些序列中的每一个都会产生问题的另一个解。还考虑了\(f(x,\cdot)\)在原点为负的情况,即所谓的半正负问题。我们的结果是基于一定的迭代模式,其中我们采用了由Noussair和Swanson开发的下解和上解方法。该方法允许我们考虑包含泛函系数\(g\)的对流项的超线性问题,而不需要径向对称。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Opuscula Mathematica
Opuscula Mathematica MATHEMATICS-
CiteScore
1.70
自引率
20.00%
发文量
30
审稿时长
22 weeks
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