In this paper, systems of fractional Laplacian equations are investigated, which involve critical homogeneous nonlinearities and Hardy-type \textcolor[rgb]{1.00,0.00,0.00}{terms} as follows \begin{eqnarray*} \begin{cases} &(- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{\lambda \mu}{\mu + \eta } \frac{|u|^{\mu - 2} |v|^{\eta} u}{|x|^{\alpha}}+ \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_u(u,v)}{|x|^{\alpha}}, ~ in~ \mathbb{R}^N,\\ \\ & (- \Delta)^s v - \gamma \frac{v}{|x|^{2 s}} = \frac{\lambda \eta}{\mu + \eta } \frac{|u|^{\mu} |v|^{\eta - 2} v}{|x|^{\alpha}} + \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_v(u,v)}{|x|^{\alpha}}, ~ in ~\mathbb{R}^N, \end{cases} \end{eqnarray*} where $0 < s < 1$, $0< \alpha < 2s <N$, $0 < \gamma < \gamma_H$ with $\gamma_H =4^s \frac{\Gamma^2(\frac{N+2s}{4})}{\Gamma^2(\frac{N-2s}{4})}, \;\;\; \textrm{for} \; i = 1,\ldots,l$ being the fractional best Hardy constant on $\mathbb{R}^N$, $2^*_s (\alpha) = \frac{2 (N -\alpha)}{N - 2 s}$ is critical Hardy-Sobolev exponent, $\mu, \eta > 1$ with $\mu + \eta = 2^*_s(\alpha)$, $Q$ is $G$-symmetric functions (${G}$ is a closed subgroup of $O(N)$, see Section 2 for details) satisfying some appropriate conditions which will be specified later, $\lambda$ is real parameter, $H_u$, $H_v$ are the partial derivative of the 2-variable $C^1$-functions $H (u,v)$ and $(-\Delta)^s$ is the fractional Laplacian operator which (up to normalization factors) may be defined as \begin{equation*} (- \Delta)^s u (x) = - \frac12\int_ {\mathbb{R}^{N} }         \frac{u (x + y) + u (x - y) - 2 u (x) }{|x-y|^{N + 2 s}} dy. \end{equation*} By variational methods and local compactness of Palais-Smale sequences, the extremals of the corresponding best Hardy-Sobolev constant are found and the existence of solutions to the system is established.\\

### Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

#### Abstract

In this paper, systems of fractional Laplacian equations are investigated, which involve critical homogeneous nonlinearities and Hardy-type \textcolor[rgb]{1.00,0.00,0.00}{terms} as follows \begin{eqnarray*} \begin{cases} &(- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{\lambda \mu}{\mu + \eta } \frac{|u|^{\mu - 2} |v|^{\eta} u}{|x|^{\alpha}}+ \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_u(u,v)}{|x|^{\alpha}}, ~ in~ \mathbb{R}^N,\\ \\ & (- \Delta)^s v - \gamma \frac{v}{|x|^{2 s}} = \frac{\lambda \eta}{\mu + \eta } \frac{|u|^{\mu} |v|^{\eta - 2} v}{|x|^{\alpha}} + \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_v(u,v)}{|x|^{\alpha}}, ~ in ~\mathbb{R}^N, \end{cases} \end{eqnarray*} where $0 < s < 1$, $0< \alpha < 2s 1$ with $\mu + \eta = 2^*_s(\alpha)$, $Q$ is $G$-symmetric functions (${G}$ is a closed subgroup of $O(N)$, see Section 2 for details) satisfying some appropriate conditions which will be specified later, $\lambda$ is real parameter, $H_u$, $H_v$ are the partial derivative of the 2-variable $C^1$-functions $H (u,v)$ and $(-\Delta)^s$ is the fractional Laplacian operator which (up to normalization factors) may be defined as \begin{equation*} (- \Delta)^s u (x) = - \frac12\int_ {\mathbb{R}^{N} }         \frac{u (x + y) + u (x - y) - 2 u (x) }{|x-y|^{N + 2 s}} dy. \end{equation*} By variational methods and local compactness of Palais-Smale sequences, the extremals of the corresponding best Hardy-Sobolev constant are found and the existence of solutions to the system is established.\\
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2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/265051
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