论文标题
由奇异性驱动的混合顺序椭圆问题，带有关键变量指数的不连续功率非线性驱动
Mixed order elliptic problems driven by a singularity, a Choquard type term and a discontinuous power nonlinearity with critical variable exponents
论文作者
论文摘要
我们证明了具有以下关键的Choquard类型问题的解决方案，该问题具有可变的分数laplacian和可变的单数指数\ begin {align*} \ begin {split} a（-Δ）^{s（\ cdot）} u+b（-Δ） 和 +ηH（u-α）| u |^{r（x）-2} u，〜\ text {in} 〜Ω， u＆= 0，〜\ text {in}〜\ mathbb {r}^n \setMinusΩ。 \ end {split} \ end {align*}其中$ a（-Δ）^{s（\ cdot）}+b（-Δ）$是具有变量订单$ s（\ cdot）的混合操作员：\ m athbb {r}^{2n}^{2n} \ rightArrow（0,1） heaviside函数（即$ h（t）= 0 $如果$ t \ leq0 $，$ h（t）= 1 $如果$ t> 0），$ $ $ ch \ subset \ subset \ mathbb {r}^n $是一个有界域，$ n \ geq 2 $，$ veq 2 $，$λ> 0 $，$λ> 0 $，$，$λ> 0 $，， $ 0 <γ^{ - } = \ underSet {x \ in \barΩ} {\ inf} \ {γ（x）\} \leqγ（x）\leqγ^+= \ = \ unterSet {x \ in \barΩ}参数和$ f $是合适的$ f $的原始功能。变量指数$ r（x）$可以等于关键指数$ 2_ {s}^*（x）= \ frac {2n} {n-2 \ bar {s}（x）} $，带有$ \ bar {s}（s}（s}（x）= s（x）= s（x）= s（x，x）$ for Some $ x \ in \ bar bar \ bar $ bar $ suitty as $ as $ as $ s a a $ s a a a a a a a a a $ as A我们还表明，作为$α\ rightarrow 0^+$，相应的解决方案将$α= 0 $的解决方案收敛到上述问题的解决方案。
We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{align*} where $a(-Δ)^{s(\cdot)}+b(-Δ)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(t) = 1$ if $t>0),$ $Ω\subset\mathbb{R}^N$ is a bounded domain, $N\geq 2$, $λ>0$, $0<γ^{-}=\underset{x\in\barΩ}{\inf}\{γ(x)\}\leqγ(x)\leqγ^+=\underset{x\in\barΩ}{\sup}\{γ(x)\}<1$, $μ$ is a continuous variable parameter, and $F$ is the primitive function of a suitable $f$. The variable exponent $r(x)$ can be equal to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$ with $\bar{s}(x)=s(x,x)$ for some $x\in\barΩ,$ and $η$ is a positive parameter. We also show that as $α\rightarrow 0^+$, the corresponding solution converges to a solution for the above problem with $α=0$.
