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