Schrödinger-Hardy system without Ambrosetti-Rabinowitz condition on Carnot groups
In this paper, we study the following Schrödinger–Hardy system −∆Gu − µ 2 r(ξ) 2 u = Fu(ξ, u, v) in Ω, −∆Gv − ν 2 r(ξ) 2 v = Fv(ξ, u, v) in Ω, u = v = 0 on ∂Ω, where Ω is a smooth bounded domain on Carnot groups G, whose homogeneous dimension is Q ≥ 3, ∆G denotes the sub-Laplacian operator on G, µ a...
Elmentve itt :
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2024
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| Sorozat: | Electronic journal of qualitative theory of differential equations
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| Kulcsszavak: | Schrödinger-Hardy rendszer, Differenciálegyenlet - nemlineáris - elliptikus - parciális |
| Tárgyszavak: | |
| doi: | 10.14232/ejqtde.2024.1.23 |
| Online Access: | http://acta.bibl.u-szeged.hu/88825 |
| Tartalmi kivonat: | In this paper, we study the following Schrödinger–Hardy system −∆Gu − µ 2 r(ξ) 2 u = Fu(ξ, u, v) in Ω, −∆Gv − ν 2 r(ξ) 2 v = Fv(ξ, u, v) in Ω, u = v = 0 on ∂Ω, where Ω is a smooth bounded domain on Carnot groups G, whose homogeneous dimension is Q ≥ 3, ∆G denotes the sub-Laplacian operator on G, µ and ν are real parameters, r(ξ) is the natural gauge associated with fundamental solution of −∆G on G, ψ is the geometrical function defined as ψ = |∇Gr|, and ∇G is the horizontal gradient associated with ∆G. The difficulty is not only the nonlinearities Fu and Fv without Ambrosetti–Rabinowitz condition, but also the hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework. |
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| Terjedelem/Fizikai jellemzők: | 21 |
| ISSN: | 1417-3875 |