Existence of solutions for a fractional elliptic problem with critical Sobolev-Hardy nonlinearities in R^N
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<...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/12/abstr.html |
| Summary: | 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 $a(x),k(x)$. |
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| ISSN: | 1072-6691 |