论文标题
由非本地诺伊曼条件的奇异性驱动的外部域中的椭圆问题
Elliptic problem in an exterior domain driven by a singularity with a nonlocal Neumann condition
论文作者
论文摘要
我们证明了解决以下问题的基础解决方案。开始{align*}（-δ）^{s} u+u＆=λ| u |^{ - γ-1} u+p（x）| u |^{p-1} u，〜\ \ \ text {in}〜\ mathbb {r} n_su（x）＆= 0，〜\ text {in} 〜Ω \ end {align*}其中$ n \ geq2 $，$λ> 0 $，$ 0 <s，γ<1 $，$ p \ in（1,2_s^*-1）$，带有$ 2_S^*= \ frac*= \ frac*= \ frac*\ freac {2n} n-2n} ％$ 0 <s^ - = \ unterSet {（x，y）\inΩ\limsΩ} {\ inf} \ {s（x，x，y）\} \ leq s（x，x，y）\ leq s^+= \ = \ nideset $ 0 <γ^ - = \ udset {x \inΩ} {\ inf} \ {γ（x）\} \ leqγ（x）\ leqleqγ^+= \ underSet {x \inΩ}} $1-γ^ - <1 <p^ - = \ usterset {x \inΩ} {\ inf} \ {p（x）\} \ leq p（x）\ leq p^+= \ unterSet {x \inΩ} {\ sup} \ {p（x）\} <2_ {s^ - }^*= \ unterSet {x \inΩ} {\inΩ} {\ incy} {\ incy} {\ incy} \ {2_s^*（x）\} $ with with $ 2_s^*（x）= \ frac {2n} {n-2 \ tilde {s}（s}（s）} $，其中$ \ tilde {s}（x）（x）= s（x，x，x）$。 此外，$ω\ subset \ mathbb {r}^n $是一个平稳的界面域，$（ - δ）^s $表示$ s $ s $ frastional laplacian，最后$ n_s $表示非局部操作员描述了neumann边界条件，如下所述。 \ begin {align*} n_ {s} u（x）＆= c_ {n，s} \ int _ {\ mathbb {r}^n \setMinusΩ} \ frac {u（x）-u（x）-u（x）-u（x）-u（x）-u（x）-u（y）} {| x-y | x-y |^n+2s} { \ end {Align*}我们进一步建立了无限的许多有限解决方案。
We prove the existence of ground state solution to the following problem. \begin{align*} (-Δ)^{s}u+u&=λ|u|^{-γ-1}u+P(x)|u|^{p-1}u,~\text{in}~\mathbb{R}^N\setminusΩ\\ N_su(x)&=0,~\text{in}~Ω\end{align*} where $N\geq2$, $λ>0$, $0<s,γ<1$, $p\in(1,2_s^*-1)$ with $2_s^*=\frac{2N}{N-2s}$. % $0<s^-=\underset{(x,y)\inΩ\timesΩ}{\inf}\{s(x,y)\}\leq s(x,y)\leq s^+=\underset{(x,y)\inΩ\timesΩ}{\sup}\{s(x,y)\}<1$, $0<γ^-=\underset{x\inΩ}{\inf}\{γ(x)\}\leq γ(x)\leq γ^+=\underset{x\inΩ}{\sup}\{γ(x)\}<1$, $1-γ^-<1<p^-=\underset{x\inΩ}{\inf}\{p(x)\}\leq p(x)\leq p^+=\underset{x\inΩ}{\sup}\{p(x)\}<2_{s^-}^*=\underset{x\inΩ}{\inf}\{2_s^*(x)\}$ with $2_s^*(x)=\frac{2N}{N-2\tilde{s}(s)}$ where $\tilde{s}(x)=s(x,x)$. Moreover, $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $(-Δ)^s$ denotes the $s$-fractional Laplacian and finally $N_s$ denotes the nonlocal operator that describes the Neumann boundary condition which is given as follows. \begin{align*} N_{s}u(x)&=C_{N,s}\int_{\mathbb{R}^N\setminusΩ}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy,~x\inΩ. \end{align*} We further establish the existence of infinitely many bounded solutions to the problem.
