Existence and Multiplicity of Solutions to a Kirchhoff Type Elliptic System with Trudinger–Moser Growth
Existence and Multiplicity of Solutions to a Kirchhoff Type Elliptic System with Trudinger–Moser Growth
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DOI：
10.1007/s00025-022-01763-9
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发表时间：
2022-01
影响因子：
2.2
通讯作者：
Shengbing Deng;X. Tian
中科院分区：
文献类型：
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作者：
Shengbing Deng;X. Tian
研究一类具有Trudinger-Moser指数增长的非线性项的Kirchhoff型椭圆方程组解的存在性和多解性。本文首先研究如下方程组解的存在性：-（a1 + B <$u <$2（θ 1-1））Δ u= λ Hu（x，u，v）in Ω，-（a2 + B <$v <$2（θ 2-1））Δ v= λ Hv（x，u，v）in Ω，u= 0，v= 0 on Ω，其中Ω是R2中具有光滑边界的有界区域，Ω w <$=（<$Ω| w| 2dx）1/2，Hu（x，u，v）和Hv（x，u，v）的行为类似于e β|（u，v）|2当|（u，v）|当β> 0，a1，a2> 0，B 1，B 2> 0，θ 1，θ 2> 1且λ为正参数时，→∞.在本文的后半部分，我们还讨论了上述系统的一个新的多重性结果，该系统具有一个由非局部依赖引起的正参数。由于Trudinger-Moser嵌入，Kirchhoff项和相关能量泛函缺乏紧致性必须通过一些新技术来克服。
This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic systems involving nonlinearities with Trudinger-Moser exponential growth. We first study the existence of solutions for the following system:-(a 1+ b 1‖ u‖ 2 (θ 1-1)) Δ u= λ H u (x, u, v) in Ω,-(a 2+ b 2‖ v‖ 2 (θ 2-1)) Δ v= λ H v (x, u, v) in Ω, u= 0, v= 0 on∂ Ω, where Ω is a bounded domain in R 2 with smooth boundary,‖ w‖=(∫ Ω|∇ w| 2 dx) 1/2, H u (x, u, v) and H v (x, u, v) behave like e β|(u, v)| 2 when|(u, v)|→∞ for some β> 0, a 1, a 2> 0, b 1, b 2> 0, θ 1, θ 2> 1 and λ is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.