Abstract
In this article, we study the following critical problem involving the fractional Laplacian: $$\begin{cases}(-\Delta)^{\frac{\alpha}{2}}u-\gamma\frac{u}{|x|^\alpha}=\lambda\frac{|u|^{q-2}}{|x|^s}+\frac{u^{2_\alpha^*(t)-2}u}{|x|^t} & in\;\Omega, \\u=0 & in\;\mathbb{R}^N\backslash\Omega,\end{cases}$$ where Ω ⊂ ℝN (N > α) is a bounded smooth domain containing the origin, α ∈ (0,2), 0 ≤ s, t 0, $$2_\alpha^*(t)=\frac{2(N-t)}{N-\alpha}$$ is the fractional critical Sobolev-Hardy exponent, 0 ≤ γ < γH, and γH is the sharp constant of the Sobolev-Hardy inequality. We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques.
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