Fractional Hardy-Sobolev equations with nonhomogeneous terms

Abstract

Abstract This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: ( − Δ ) s u − γ u | x | 2 s = K ( x ) | u | 2 s ∗ ( t ) − 2 u | x | t + f ( x ) in R N , u ∈ H ˙ s ( R N ) , $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb R^N), \end{cases} \end{array}$$ where N > 2s, s ∈ (0, 1), 0 ≤ t < 2s < N and 2 s ∗ ( t ) := 2 ( N − t ) N − 2 s $\begin{array}{} \displaystyle 2^*_s(t):=\frac{2(N-t)}{N-2s} \end{array}$ . 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 Ḣs (ℝ N )′ of Ḣs (ℝ N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and ∥f∥(Ḣs )′ is small enough (but f ≢ 0), we establish existence of at least two positive solutions to the above equation.