# A Global Compact Result for a Fractional Elliptic Problem with Critical Sobolev-Hardy Nonlinearities on ${\mathbb R}^N$

@article{Jin2017AGC, title={A Global Compact Result for a Fractional Elliptic Problem with Critical Sobolev-Hardy Nonlinearities on \$\{\mathbb R\}^N\$}, author={Lingyu Jin and Shaomei Fang}, journal={arXiv: Analysis of PDEs}, year={2017} }

In this paper, we are concerned with the following type of elliptic problems: $$
(-\Delta)^{\alpha} u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^\alpha({\mathbb R}^N), $$ where $2<q< 2^*$, $0<\alpha<1$, $0<s<2\alpha$, $2^*_{s}=2(N-s)/(N-2\alpha)$ 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 to the problem, we obtain the existence…

## References

SHOWING 1-10 OF 27 REFERENCES

### Fractional Laplacian equations with critical Sobolev exponent

- Mathematics
- 2015

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$LK$$\begin{aligned} \left\{ \begin{array}{l@{\quad…

### Weighted Fractional Sobolev Inequality in ℝN

- Mathematics
- 2016

Abstract In this paper, we show that the minimizing problem Λ s , N , k , α = inf u ∈ H ˙ s ( ℝ N ) , u ≢ 0 ∫ ℝ N | ( - Δ ) s 2 u ( x ) | 2 𝑑 x ( ∫ ℝ N | u ( x ) | 2 s , α * | y | α 𝑑…

### Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

- Mathematics
- 2013

In this paper, we study the following problem
\begin{eqnarray}
(-\Delta)^{\frac{\alpha}{2}}u = K(x)|u|^{2_{\alpha}^{*}-2}u
+ f(x) \quad in \ \Omega,\\
u=0 \quad on \ \partial \Omega, …

### A Robin boundary problem with Hardy potential and critical nonlinearities

- Mathematics
- 2008

AbstractLet Ω be a bounded domain with a smooth C2 boundary in ℝn (n ≥ 3), 0 ∈
$$\bar \Omega $$
, and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class…

### Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2012

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional Laplacian…

### Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces

- Mathematics
- 2013

We obtain an improved Sobolev inequality in $$\dot{H}^s$$H˙s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of…

### BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL NONLOCAL FRACTIONAL LAPLACIAN PROBLEMS

- Mathematics
- 2016

### A global compactness result for singular elliptic problems involving critical Sobolev exponent

- Mathematics
- 2002

Let Ω C R N be a bounded domain such that 0 ∈ Ω, N > 3, 2* = 2N/N*2,, λ ∈ R, ∈∈ R. Let {u n } C H 1 0,(Ω) be a (P.S.) sequence of the functional E λ, ∈(u) = 1/2 ∫ Ω (|⊇u| 2 - λu 2 /|x| 2 - ∈u 2 ) - ∫…

### The Brezis-Nirenberg result for the fractional Laplacian

- Mathematics
- 2014

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983).…