Electron. J. Differential Equations, Vol. 2018 (2018), No. 12, pp. 1-23.

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

Lingyu Jin, Shaomei Fang

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 $2<q< 2^*$, $0<\alpha<1$, $N>4\alpha$, $0<s<2\alpha$, $2^*_{s}=2(N-s)/(N-2\alpha)$ 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 $a(x),k(x)$.

Submitted April 8, 2017. Published January 10, 2018.
Math Subject Classifications: 35J10, 35J20, 35J60.
Key Words: Fractional Laplacian; compactness; positive solution; unbounded domain; Sobolev-Hardy nonlinearity.

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Lingyu Jin
Department of Mathematics
South China Agricultural University
Guangzhou 510642, China
email: jinlingyu300@126.com
Shaomei Fang
Department of Mathematics
South China Agricultural University
Guangzhou 510642, China
email: fangshaomeidz90@126.com

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