论文标题
关键的生长分数Kirchhoff椭圆问题
Critical growth fractional Kirchhoff elliptic problems
论文作者
论文摘要
本文涉及以下分数Kirchhoff-Choquard问题的积极弱解决方案的存在和多样性： \ begin {equation*} \ begin {array} {cc} \ \ displayStyle m \ left（\ | u \ |^2 \右）（-Δ）^s u = \dsλf（x）| u |^{q-2} u + + \ left（\ int \limits_Ω\ frac {| frac {| dy \ right）| u |^{2^{*} _ {μ，s} -2} u \; \ text {in} \; ω， u> 0 \ quad \ text {in} \; ω，\，\， u = 0 \ quad \ text {in} \; \ Mathbb {r}^{n} \backslashΩ， \ end {array} \ end {equation*} 其中$ω$是$ \ mathbb {r}^{n} $的打开界域，带有$ c^2 $边界，$ n> 2s $和$ s \ in（0,1）$，$ m $ model型kirchhoff-type系数的形式$ m（t）= a + bt^a + bt^a + bt^{ $（ - δ）^s $是分数拉普拉斯操作员，$λ> 0 $是一个真正的参数。我们使用变分方法探索$ {q} \ in（1,2^*_ s）$和$ \ te \ geq 1 $的解决方案的存在。 ％，我们还考虑$ \ te> 2^*_ {μ，s} $的情况，价格为$ 2 <q <2^*_ {s} $。 这里$ 2^*_ s = \ frac {2n} {n-2s} $和$ 2^{*} _ {μ，s} = \ frac {2n-μ} {n-2s} $是顽固的witewwood-sobolev noreality的关键指数。
This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-Δ)^s u = \dsλf(x)|u|^{q-2}u + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, u > 0\quad \text{in} \; Ω, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\te-1}$, where $a, b > 0$ are given constants. $(-Δ)^s$ is fractional Laplace operator, $λ> 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{μ,s}$ for $2< q < 2^*_{s}$. Here $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.
