论文标题
至少存在$ k $解决方案涉及奇异性和关键指数的分数$ p $ -kirchhoff问题
Existence of at least $k$ solutions to a fractional $p$-Kirchhoff problem involving singularity and critical exponent
论文作者
论文摘要
我们研究了以下非局部椭圆形问题的非负溶液的存在，涉及奇异性 \ begin {align} \ Mathfrak {m} \ left（\ int_ {q} \ frac {| u（x）-u（y）-u（y）|^p} {| x-y |^|^{n+ps}} dxdy \ right） u＆= \fracλ{| u |^{γ-1} u}+| u |^{p_s^* - 2} u〜 \ text {in} 〜Ω，\ nonumber u＆> 0〜 \ text {in} 〜Ω，\ nonumber u＆= 0〜 \ text {in}〜\ mathbb {r}^n \setMinusΩ，\ nonumber \ end {align} 其中$ω\ subset \ mathbb {r}^n $，是一个具有Lipschitz边界的有界域，$λ> 0 $，$ n> ps $，$ 0 <s，γ<1 $，$（Δ）_ {p}^{p}^{s}^{s}^{s}^{s} $ p_s^*= \ frac {np} {n-ps} $是关键的sobolev endent。我们采用{\ it cut-off}参数来获得$ k $（任意大整数）解决方案的存在。此外，通过使用Moser迭代技术，我们证明了一个均匀的$ l^{\ infty}（ω）$用于解决方案。这项工作的新颖性在于，尽管存在关键的非线性术语，但通过使用对称山间定理证明了小型解决方案的存在，当然，这当然是超级线性的。
We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity \begin{align} \mathfrak{M}\left(\int_{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-Δ)_{p}^{s} u&=\fracλ{|u|^{γ-1}u}+|u|^{p_s^*-2}u~\text{in}~Ω,\nonumber u&>0~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N$, is a bounded domain with Lipschitz boundary, $λ>0$, $N>ps$, $0<s,γ<1$, $(-Δ)_{p}^{s}$ is the fractional $p$-Laplacian operator for $1<p<\infty$ and $p_s^*=\frac{Np}{N-ps}$ is the critical Sobolev exponent. We employ a {\it cut-off} argument to obtain the existence of $k$ (being an arbitrarily large integer) solutions. Furthermore, by using the Moser iteration technique, we prove a uniform $L^{\infty}(Ω)$ bound for the solutions. The novelty of this work lies in proving the existence of small energy solutions by using the symmetric mountain pass theorem in spite of the presence of a critical nonlinear term which, of course, is super-linear.
