Existence of solutions for a fractional elliptic problem with critical Sobolev-Hardy nonlinearities in R^N

Abstract

In this article, we study the fractional elliptic equation with critical Sobolev-Hardy nonlinearity $$\displaylines{ (-\Delta)^{\alpha} u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u,\cr u\in H^\alpha(\mathbb{R}^N), }$$ where $24\alpha$, $0<s<2\alpha$, $2^*_{s}=2(N-s)/(N-2\alpha)$ is the critical Sobolev-Hardy exponent, $2^*=2N/(N-2\alpha)$ is the critical Sobolev exponent, $a(x),k(x)\in C(\mathbb{R}^N)$. Through a compactness analysis of the functional associated, we obtain the existence of positive solutions under certain assumptions on $a(x),k(x)$.

Keywords

• Fractional Laplacian
• compactness
• positive solution
• unbounded domain
• Sobolev-Hardy nonlinearity