#### Shahla Amiri, Nemat Nyamoradi, Abolfazl Behzadi, Multiple solutions for Hardy nonlocal fractional elliptic equations in $R^N$, Vol. 2019 (2019), Article ID 19, pp. 1-12

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DOI: 10.23952/jnfa.2019.19

Received December 5, 2018; Accepted May 3, 2019; Published May 14, 2019

Abstract. In this paper, we use Ekeland’s variational principle to study the existence of at least three nontrivial solutions for the following critical nonlocal fractional Hardy elliptic equation $(- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}}= \frac{|u|^{2^*_s (\alpha)- 2}u}{|x|^{\alpha}}+\lambda f(x,u)$ in $\mathbb{R}^N,$ where $N > 2s$, $0 < s < 1$, $\gamma,\lambda$ are real parameters, $2^*_s (\alpha) = \frac{2 (N -\alpha)}{N - 2 s}$ is critical Hardy-Sobolev exponent with $\alpha \in [0, 2s)$, $f:\mathbb{R}^{N}\times \mathbb{R} \to \mathbb{R}$ is a suitable function and $(- \Delta)^s$ is the fractional Laplace operator.

Shahla Amiri, Nemat Nyamoradi, Abolfazl Behzadi, Multiple solutions for Hardy nonlocal fractional elliptic equations in $\mathbb{R}^N$, Journal of Nonlinear Functional Analysis, Vol. 2019 (2019), Article ID 19, pp. 1-12