论文标题
非本地$ p $ -kirchhoff方程，具有单数和关键的非线性项
Nonlocal $p$-Kirchhoff equations with singular and critical nonlinearity terms
论文作者
论文摘要
这项工作的目的是研究一个非局部问题，涉及奇异和关键的非线性：\ begin {equination*} \ left \ {\ begin {array} {ll} {ll}（[u] _ {s，p}，p}^p） p_s^{*} -1} \ quad \ text {in}ω，\\ u> 0，\; \; \; \; \; \ quad \ quad \ text {in}ω，\\ u = 0，\ u = 0，\; \; \; \; \; \; \; \ quad; ω，\ end {array} \ right。 \ end {equation*}其中$ω$是$ \ mathbb {r}^n $中的一个有界域，带有光滑的边界$ \ partialω$，$ 0 <s <s <1 <p <\ p <\ infty $，$ n> sp $，$ 1 <p^p^p^*s/p，$ p^*_ s/p，$ （ - δ）_p^s $是非局部$ p $ -laplace操作员，$ [u] _ {s，p} $是gagliardo $ p $ -seminorm。我们将一些变分技术与截断参数相结合，以表明上述问题的积极解决方案的存在和多样性。
The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{σ-1}(-Δ)^s_p u = \fracλ{u^γ}+u^{ p_s^{*}-1 }\quad \text{in }Ω,\\ u>0,\;\;\;\;\quad \text{in }Ω,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus Ω,\end{array} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with the smooth boundary $\partial Ω$, $0 < s< 1<p<\infty$, $N> sp$, $1<σ<p^*_s/p,$ with $p_s^{*}=\frac{Np}{N-ps},$ $ (- Δ)_p^s$ is the nonlocal $p$-Laplace operator and $[u]_{s,p}$ is the Gagliardo $p$-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.
