Electron. J. Differential Equations, Vol. 2019 (2019), No. 54, pp. 1-19.

Existence of solutions for Kirchhoff-type problems via the method of lower and upper solutions

Baoqiang Yan, Donal O'Regan, Ravi P. Agarwal

This article considers elliptic problems of Kirchhoff-type. We give some new definitions of lower and upper solutions for the problem and establish the method of lower and upper solutions when the upper and lower solutions are well ordered, i.e., the lower solution is less than the upper one, and we also consider the case when the upper and lower solutions have opposite ordering. In addition we use the relation between the topological degree and strict upper and lower solutions in both cases and using this we obtain multiplicity results for nonlinear Kirchhoff-type elliptic problems.

Submitted January 31, 2019. Published April 29, 2019.
Math Subject Classifications: 35J60, 35J75, 47H10.
Key Words: Kirchhoff-type elliptic problem; lower and upper solution; topological degree.

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Baoqiang Yan
School of Mathematical Sciences
Shandong Normal University
Jinan, 250014, China
email: yanbqcn@aliyun.com
Donal O'Regan
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland
Galway, Ireland
email: donal.oregan@nuigalway.ie
Ravi P. Agarwal
Department of Mathematics
Texas A and M University-Kingsville
Texas 78363, USA
email: Ravi.Agarwal@tamuk.edu

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