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Existence of solutions to Kirchhoff type equations involving the nonlocal p1& … &pm fractional Laplacian with critical Sobolev-Hardy exponent
Complex Variables and Elliptic Equations ( IF 0.9 ) Pub Date : 2021-04-28 , DOI: 10.1080/17476933.2021.1913129
Wei Chen 1 , Nguyen Van Thin 2, 3
Affiliation  

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p1&&pm fractional Laplacian with critical Sobolev-Hardy exponent M1R2N|u(x)u(y)|p1|xy|N+p1s1dxdy(Δ)p1s1u++MmR2N|u(x)u(y)|pm|xy|N+pmsmdxdy(Δ)pmsmu=f(x,u)+γ|u|ps(α)2u|x|α+β|u|q2u|x|αin Ω,u=0in RNΩ, where 0<sm<<s1=s<1, 1<pmp1=p<Ns, m1, β,γ are nonnegative constants and ps(α)=p(Nα)Nspps(0) is called the critical Sobolev-Hardy exponent, 1<q<p, 0α<ps. Here (Δ)rs, with r{p1,,pm} is the fractional r-Laplace operator. Ω is an open bounded subset of RN with smooth boundary and 0Ω. M1,,Mm are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when γ=0. We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, we consider the case N=ps, and study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.



中文翻译:

涉及具有临界 Sobolev-Hardy 指数的非局部 p1& … &pm 分数拉普拉斯算子的 Kirchhoff 型方程的解的存在性

本文的目的是研究涉及非局部的 Kirchhoff 型方程的解的存在性。p1&&p具有临界 Sobolev-Hardy 指数的分数拉普拉斯算子1R2ñ|(X)-(是的)|p1|X-是的|ñ+p1s1dXd是的(-Δ)p1s1++R2ñ|(X)-(是的)|p|X-是的|ñ+psdXd是的(-Δ)ps=F(X,)+γ||ps*(α)-2|X|α+β||q-2|X|α一世n Ω,=0一世n RñΩ,在哪里0<s<<s1=s<1, 1<pp1=p<ñs, 1, β,γ是非负常数和ps*(α)=p(ñ-α)ñ-spps*(0)称为临界 Sobolev-Hardy 指数,1<q<p, 0α<ps. 这里(-Δ)rs,r{p1,,p}是分数r -Laplace 算子。Ω 是一个开有界子集Rñ具有光滑的边界和0Ω.1,,是连续函数,f是不满足 Ambrosetti-Rabinowitz 条件的 Carathéodory 函数。通过使用山口定理,我们得到了上述问题的解的存在性。此外,使用喷泉定理,我们得到上述问题的无限多解的存在性,当γ=0. 我们还通过 Morse 理论研究了涉及分数p-拉普拉斯算子的 Kirchhoff 型方程的两个非平凡解的存在。最后,我们考虑这种情况ñ=ps,并研究涉及 Trudinger-Moser 非线性的退化 Kirchhoff 方程。据我们所知,这是我们第一次在这个领域研究我们的问题。

更新日期:2021-04-28
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