On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

Abstract

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:where s ∈ (0, 1), N > 2s, 2 * s := 2N N−2s , 0 < a ∈ L ∞ (R N ) and f is a nonnegative nontrivial functional in the dual space ofḢ s i.e., (Ḣ s ) ′ f, u Ḣs ≥ 0, whenever u is a nonnegative function inḢ s . We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that a is a continuous function, a(x) ≥ 1 in R N , a(x) → 1 as |x| → ∞ and f Ḣs (R N ) ′ is small enough (but f ≡ 0), we establish existence of at least two positive solutions to (E). 2010 MSC: 35R11, 35A15, 35B33, 35J60 LetḢ s (R N ) := u ∈ L 2 * s (R N ) :¨R 2N |u(x) − u(y)| 2 |x − y| N +2s dx dy < ∞ , be the homogeneous fractional Sobolev space, endowed with the inner product ·, · Ḣs and corresponding Gagliardo norm u Ḣs (R N ) := ¨R 2N |u(x) − u(y)| 2 |x − y| N +2s dx dy

Fractional Hardy-Sobolev equations with nonhomogeneous terms

“…The main purpose of this section is to classify (P S) sequences of the functional Īγ K,t,f . Classification of (P S) sequences has been done for various problems having lack of compactness, to quote a few, we cite [7,20,21] in the nonlocal case with γ = 0 = t, while in the local case [8,24] with Hardy potentials and in [25] without Hardy potentials. We also refer to [26] for a more abstract approach of the profile decomposition in general Hilbert spaces.…”

“…We establish a classification theorem for the (P S) sequences of (2.1) in the spirit of the above results. In [7,20], the noncompactness is completely described by the single blow up profile W , which is a solution of…”

“…We are going to prove existence and multiplicity of positive solutions of (E γ K,t,f ) in the spirit of [6,7]. Under the conditions on K and f stated above, equation (E γ K,t,f ) can be regarded as a perturbation problem of the homogeneous equation (E γ 1,t,0 ).…”

Fractional elliptic systems with critical nonlinearities

“…To overcome this, it is necessary to look for a nice energy range where the (P S) condition holds in order to use variational arguments. Classification of (P S) sequences associated with a scalar equation (local/nonlocal) has been done in many papers, to quote a few, we cite [4,20,31,33,34,38]. To the best of our knowledge, the (P S) decomposition associated to systems of equations has not been studied much.…”

“…where α, β, f, g are as in (S) and the potentials a, b are continuous functions in R N with a, b ≥ 1 and a(x), b(x) → 1 as |x| → ∞. See for instance [4] in the scalar case. c) One can also try to adopt the methodology of this paper in order to study the system of equations involving the Hardy operator i.e., if (−∆) s is replaced by the Hardy operator (−∆) s − γ |x| 2s , where γ ∈ (0, γ N,s ) and γ N,s is the best Hardy constant in the fractional Hardy inequality.…”