Existence of Solutions to Fractional Elliptic Equation with the Hardy Potential and Concave–Convex Nonlinearities

Mediterranean Journal of Mathematics - Tập 20 - Trang 1-16 - 2022
Xiangrui Li1, Shuibo Huang1, Qiaoyu Tian1
1School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, People’s Republic of China

Tóm tắt

In this paper, by a new variational principle established by Moameni, we establish the existence of solutions to the following fractional elliptic equation with the Hardy potential and concave–convex nonlinearities, $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u(x)- \lambda \frac{u(x)}{|x|^{2s}}=u(x)^{p-1}+\mu u(x)^{q-1}, &{}{} x\in \Omega ,\\ ~~~~~~~~~~~~~~~~~~~~~~~~~u(x)>0, &{}{}x\in \Omega ,\\ ~~~~~~~~~~~~~~~~~~~~~~~~~u(x)=0,&{}{} x\in \mathbb {R}^{N}\setminus \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ is a bounded Lipschitz domain with $$0\in \Omega $$ , $$(-\Delta )^{s}$$ is the restricted fractional Laplacian, $$02s$$ , $$0<\lambda <\Lambda _{N,s}$$ , $$\Lambda _{N,s}$$ is the sharp constant of the Hardy–Sobolev inequality, $$\begin{aligned} 1

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