This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: (− Δ )su− γ u|x|2s=K(x)|u|2s∗ (t)− 2u|x|t+f(x)inRN,u∈ H˙ s(RN), $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb R^N), \end{cases} \end{array}$$ where N > 2 s , s ∈ (0, 1), 0 ≤ t < 2 s < N and 2s∗ (t):=2(N− t)N− 2s $\begin{array}{} \displaystyle 2^*_s(t):=\frac{2(N-t)}{N-2s} \end{array}$. Here 0 < γ < γ N , s and γ N , s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on ℝ N , with K (0) = 1 = lim | x |→∞ K ( x ). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣ s (ℝ N )′ of Ḣ s (ℝ N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and ∥ f ∥ ( Ḣ s )′ is small enough (but f ≢ 0), we establish existence of at least two positive solutions to the above equation.

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具有非齐次项的分数阶Hardy-Sobolev方程

本文讨论了以下具有临界非线性的非局部方程组的正解的存在性和多重性：（-Δ）su-γu | x | 2s = K（x）| u | 2s ∗（t）− 2u | x | t + f（x）inRN，u∈H˙s（RN），$$ \ begin {array} {} \ displaystyle \ begin {cases}（-{\ it \ Delta}）^ su-\ gamma \ dfrac {u} {| x | ^ {2s}} = K（x）\ dfrac {| u | ^ {2 ^ * _ s（t）-2} u} {| x | ^ t} + f（x）\ quad \ mbox {in} \ quad \ mathbb R ^ N，\\ \ qquad \ qquad \ qquad \ quad u \ in \ dot {H} ^ s（\ mathbb R ^ N），\ end {cases} \ end {数组} $$其中N> 2 s，s∈（0，1），0≤t <2 s <N和2s ∗（t）：= 2（N−t）N−2s $ \ begin {array} { } \ displaystyle 2 ^ * _ s（t）：= \ frac {2（Nt）} {N-2s} \ end {array} $。在此，0 <γ<γN，s和γN，s是分数Hardy不等式中的最佳Hardy常数。系数K是ℝN上的一个正连续函数，其中K（0）= 1 = lim |。x |→∞K（x）。扰动f是Ḣs（ℝN）的对偶空间Ḣs（ℝN）'的非负非平凡函数。我们建立与功能相关的Palais-Smale序列的轮廓分解。此外，如果K≥1且f f∥（Ḣs）'足够小（但f≢0），则对上述方程式建立至少两个正解的存在。