论文标题
无限的许多小解决方案，用于具有奇异非线性的可变指数的椭圆形PDE
Infinitely many small solutions to an elliptic PDE of variable exponent with a singular nonlinearity
论文作者
论文摘要
我们证明了对以下非本地椭圆偏微分方程的无限多个非负解的存在 （-Δ）_ {p（\ cdot）}^{s} u＆= \fracλ{| u |^{γ（x）-1} u}+f（x，x，x，u）〜\ text {in} 〜Ω，\ nonumber u＆= 0〜 \ text {in}〜\ mathbb {r}^n \setMinusΩ，\ nonumber \ end {align}，其中$ω\ subset \ subset \ subset \ mathbb {r}^n，\，\，n \ geq2 $是一个平稳的，有界的，有边界的，$ c $，$λ> 0 $，$，$，$，0,1 \，（0,1）所有$ x \ in \barΩ$，$ n> sp（x，y）$的$γ（x）\ in \barΩ非线性功能$ f $满足某些增长条件。此外，我们通过Moser Iteration Technique建立了一个均匀的$ l^{\ infty}（\barΩ）$估计。
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-Δ)_{p(\cdot)}^{s} u&=\fracλ{|u|^{γ(x)-1}u}+f(x,u)~\text{in}~Ω,\nonumber u&=0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{align} where $Ω\subset\mathbb{R}^N,\, N\geq2$ is a smooth, bounded domain, $λ>0$, $s\in (0,1)$, $γ(x)\in(0,1)$ for all $x\in\barΩ$, $N>sp(x,y)$ for all $(x,y)\in\barΩ\times\barΩ$ and $(-Δ)_{p(\cdot)}^{s}$ is the fractional $p(\cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{\infty}(\barΩ)$ estimate of the solution(s) by the Moser iteration technique.
