Fractional Hardy-Sobolev equations with nonhomogeneous terms

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.

On doubly critical coupled systems involving fractional Laplacian with partial singular weight

“…Bhakta et al 11 considered the existence and multiplicity of solutions to the following nonlocal equations with critical nonlinearity: where s ∈ (0, 1), 0 ≤ β < 2 s < n , , , satisfies , and f ( x ) ≢ 0 is a nonnegative functional in the dual space …”

“…Bhakta et al 11 considered the existence and multiplicity of solutions to the following nonlocal equations with critical nonlinearity:…”

On critically coupled (s_1, s_2)-fractional system of Schrödinger equations with hardy potential

“…Thus, J = ∅. If {u n } is concentrating at points zero or infinity, we argue analogously for the functional J 1 and use the result provided by [2] (See Theorem 2.1 part (v)) to obtain the following…”

Fractional Hardy equations with critical and supercritical exponents

“…In the last few years, there are several works related to Eq. (1.7); see for instance ( [5,6,19,22,23] and references therein). Authors in [5] have studied the two-sided Green function estimate of fractional Hardy operator and the integral representation of solution.…”

“…Dipierro et al in [19] have considered f (x, u) = u 2 * s −1 and proved existence, qualitative properties and asymptotic behavior of solutions both at 0 and infinity. In [6]…”