On a Fractional $p&q$ Laplacian Problem with Critical Sobolev–Hardy Exponents

We consider the following fractional $p\&q$ Laplacian problem with critical Sobolev-Hardy exponents \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s}_{p} u + (-\Delta)^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha)-2}u}{|x|^{\alpha}}+ \lambda f(x, u) & \mbox{ in } \Omega \\ u=0 & \mbox{ in } \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{equation*} where $0<s<1$, $1\leq q<p<\frac{N}{s}$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian operator, $\lambda$ is a positive parameter, $\Omega \subset \mathbb{R}^{N}$ is an open bounded domain with smooth boundary, $0\leq \alpha <sp$, and $p^{*}_{s}(\alpha)=\frac{p(N-\alpha)}{N-sp}$ is the so called Hardy-Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya \cite{K}, we show the existence of infinitely many solutions which tend to zero provided that $\lambda$ belongs to a suitable range.