The effect of the domain topology on the number of positive solutions for an elliptic system

Giovany M. Figueiredo () and Leticia S. Silva ()
Additional contact information
Giovany M. Figueiredo: Universidade de Brasília
Leticia S. Silva: Universidade de Brasília

Partial Differential Equations and Applications, 2022, vol. 3, issue 5, 1-23

Abstract: Abstract In this paper we prove an existence result of multiple positive solutions for the following system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u= \frac{2 \alpha _\epsilon }{\alpha _\epsilon +\beta _\epsilon }|u|^{\alpha _\epsilon -2}u |v|^{\beta _\epsilon }&{} \text{ in } \Omega , \\ -\Delta v= \frac{2 \beta _\epsilon }{\alpha _\epsilon +\beta _\epsilon }|u|^{\alpha _\epsilon } |v|^{\beta _\epsilon -2 }v&{} \text{ in } \Omega , \\ u= v =0 &{} \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ - Δ u = 2 α ϵ α ϵ + β ϵ | u | α ϵ - 2 u | v | β ϵ in Ω , - Δ v = 2 β ϵ α ϵ + β ϵ | u | α ϵ | v | β ϵ - 2 v in Ω , u = v = 0 on ∂ Ω , where $$\Omega $$ Ω is a smooth and bounded domain in $${\mathbb {R}}^{N}$$ R N , $$N\ge 3$$ N ≥ 3 , $$\alpha _\epsilon = \alpha - \frac{\epsilon }{2}$$ α ϵ = α - ϵ 2 , $$\beta _\epsilon =\beta - \frac{\epsilon }{2}$$ β ϵ = β - ϵ 2 , $$\alpha , \beta >1$$ α , β > 1 and $$\alpha +\beta = 2^*$$ α + β = 2 ∗ , where $$2^{*}=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 . More specifically, we prove that, for $$\epsilon >0$$ ϵ > 0 small, the number of positive solutions is estimated below by topological invariants of the domain $$\Omega $$ Ω : the Ljusternick–Schnirelmann category.

Keywords: Gradient system; Positive solutions; Primary 35J60; Secondary 35C20; 35B33; 49J45 (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-022-00202-2 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:3:y:2022:i:5:d:10.1007_s42985-022-00202-2

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-022-00202-2

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().