Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator ΔΔu(k−1)1−(Δu(k−1))2+λμ(k)(p+1)up(k)vq+1(k)=0,k∈[2,n−1]Z,ΔΔv(k−1)1−(Δv(k−1))2+λμ(k)(q+1)up+1(k)vq(k)=0,k∈[2,n−1]Z,Δu(1)=u(n)=0=Δv(1)=v(n),\left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\...
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-07-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2022-0602 |
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| author | Liang Yongwen Chen Tianlan |
| author_facet | Liang Yongwen Chen Tianlan |
| author_sort | Liang Yongwen |
| collection | DOAJ |
| description | We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator ΔΔu(k−1)1−(Δu(k−1))2+λμ(k)(p+1)up(k)vq+1(k)=0,k∈[2,n−1]Z,ΔΔv(k−1)1−(Δv(k−1))2+λμ(k)(q+1)up+1(k)vq(k)=0,k∈[2,n−1]Z,Δu(1)=u(n)=0=Δv(1)=v(n),\left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\left(\Delta u\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(p+1){u}^{p}\left(k){v}^{q+1}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta \left[\frac{\Delta v\left(k-1)}{\sqrt{1-{\left(\Delta v\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(q+1){u}^{p+1}\left(k){v}^{q}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta u\left(1)=u\left(n)=0=\Delta v\left(1)=v\left(n),& \\ \end{array}\right. where n∈Nn\in {\mathbb{N}} with n>4n\gt 4, max{p,q}>1\max \left\{p,q\right\}\gt 1, λ>0\lambda \gt 0, Δu(k−1)=u(k)−u(k−1)\Delta u\left(k-1)=u\left(k)-u\left(k-1), and μ(k)>0\mu \left(k)\gt 0 for all k∈[2,n−1]Zk\in {\left[2,n-1]}_{{\mathbb{Z}}}. The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of λ\lambda . Our main tools are based on topological methods, critical point theory, and lower and upper solutions. |
| first_indexed | 2024-03-12T17:04:03Z |
| format | Article |
| id | doaj.art-75cd53f6ad28430f911dc02b416f7f41 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-03-12T17:04:03Z |
| publishDate | 2023-07-01 |
| publisher | De Gruyter |
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| series | Open Mathematics |
| spelling | doaj.art-75cd53f6ad28430f911dc02b416f7f412023-08-07T06:56:44ZengDe GruyterOpen Mathematics2391-54552023-07-0121113115210.1515/math-2022-0602Positive solutions for discrete Minkowski curvature systems of the Lane-Emden typeLiang Yongwen0Chen Tianlan1Lanzhou Petrochemical University of Vocational Technology, Lanzhou 730060, P. R. ChinaDepartment of Mathematics, Northwest Normal University, Lanzhou 730070, P. R. ChinaWe study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator ΔΔu(k−1)1−(Δu(k−1))2+λμ(k)(p+1)up(k)vq+1(k)=0,k∈[2,n−1]Z,ΔΔv(k−1)1−(Δv(k−1))2+λμ(k)(q+1)up+1(k)vq(k)=0,k∈[2,n−1]Z,Δu(1)=u(n)=0=Δv(1)=v(n),\left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\left(\Delta u\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(p+1){u}^{p}\left(k){v}^{q+1}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta \left[\frac{\Delta v\left(k-1)}{\sqrt{1-{\left(\Delta v\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(q+1){u}^{p+1}\left(k){v}^{q}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta u\left(1)=u\left(n)=0=\Delta v\left(1)=v\left(n),& \\ \end{array}\right. where n∈Nn\in {\mathbb{N}} with n>4n\gt 4, max{p,q}>1\max \left\{p,q\right\}\gt 1, λ>0\lambda \gt 0, Δu(k−1)=u(k)−u(k−1)\Delta u\left(k-1)=u\left(k)-u\left(k-1), and μ(k)>0\mu \left(k)\gt 0 for all k∈[2,n−1]Zk\in {\left[2,n-1]}_{{\mathbb{Z}}}. The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of λ\lambda . Our main tools are based on topological methods, critical point theory, and lower and upper solutions.https://doi.org/10.1515/math-2022-0602discrete systemsminkowski curvature operatorpositive solutionbrouwer degreelower and upper solutionscritical point theory39a1347h11 |
| spellingShingle | Liang Yongwen Chen Tianlan Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type Open Mathematics discrete systems minkowski curvature operator positive solution brouwer degree lower and upper solutions critical point theory 39a13 47h11 |
| title | Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type |
| title_full | Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type |
| title_fullStr | Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type |
| title_full_unstemmed | Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type |
| title_short | Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type |
| title_sort | positive solutions for discrete minkowski curvature systems of the lane emden type |
| topic | discrete systems minkowski curvature operator positive solution brouwer degree lower and upper solutions critical point theory 39a13 47h11 |
| url | https://doi.org/10.1515/math-2022-0602 |
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