• Corpus ID: 119713872

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

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

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

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

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

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

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

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

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

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).