Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method.

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P) (− Δ) s u = λ u γ + | u | 2 α ∗ − 2 u | x | α in Ω , u > --> 0 in Ω , u = 0 in R N ∖ Ω , $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < 2 s ∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$ , where 2 s ∗ = 2 N N − 2 s and 2 α ∗ = 2 (N − α) N − 2 s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (− Δ) s u (x) = − 1 2 ∫ R N u (x + y) + u (x − y) − 2 u (x) | y | N + 2 s d y , for all x ∈ R N. $$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P). [ABSTRACT FROM AUTHOR]