Existence of solutions to kirchhoff type equations involving the nonlocal p1 pm fractional laplacian with critical sobolev hardy exponent

Complex Variables and Elliptic Equations An International Journal ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/gcov20 Existence of solutions to Kirchhoff type equations involving the nonlocal p1& … &pm fractional Laplacian with critical Sobolev-Hardy exponent Wei Chen & Nguyen Van Thin To cite this article: Wei Chen & Nguyen Van Thin (2021): Existence of solutions to Kirchhoff type equations involving the nonlocal p1& … &pm fractional Laplacian with critical Sobolev-Hardy exponent, Complex Variables and Elliptic Equations, DOI: 10.1080/17476933.2021.1913129 To link to this article: https://doi.org/10.1080/17476933.2021.1913129 Published online: 28 Apr 2021 Submit your article to this journal Article views: 52 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gcov20 COMPLEX VARIABLES AND ELLIPTIC EQUATIONS https://doi.org/10.1080/17476933.2021.1913129 Existence of solutions to Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev-Hardy exponent Wei Chen1 and Nguyen Van Thin2,3 Chongqing University of Posts and Telecommunications, School of Science, Chongqing, People’s Republic of China; Department of Mathematics, Thai Nguyen University of Education, Thai Nguyen city, Vietnam; Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Hanoi, Vietnam ABSTRACT ARTICLE HISTORY The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev-Hardy exponent Received 20 January 2020 Accepted 26 March 2021 ⎧ |u(x) − u(y)|p1 ⎪ ⎪ dx dy (− )sp11 u + · · · ⎪M ⎪ 2N |x − y|N+p1 s1 ⎪ R ⎪ ⎪ ⎪ ⎪ |u(x) − u(y)|pm ⎪ ⎨ +Mm dx dy (− )spmm u N+pm sm 2N |x − y| R ⎪ ∗ ⎪ ⎪ |u|q−2 u |u|ps (α)−2 u ⎪ ⎪ +β in , = f (x, u) + γ ⎪ ⎪ α ⎪ |x| |x|α ⎪ ⎪ ⎩ u = in RN \ , where < sm < · · · < s1 = s < 1, < pm ≤ · · · ≤ p1 = p < Ns , m ≥ ∗ 1, β, γ are nonnegative constants and p∗s (α) = p(N−α) N−sp ≤ ps (0) is called the critical Sobolev-Hardy exponent, < q < p, ≤ α < ps Here (− )sr , with r ∈ {p1 , , pm } is the fractional r-Laplace operator is an open bounded subset of RN with smooth boundary and ∈ M1 , , Mm are continuous functions and f is a Carathéodory function which does not satisfy the AmbrosettiRabinowitz condition By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when the γ = We also study the existence of two nontrivial solutions for Kirchhoff type equation involving the fractional p-Laplacian via Morse theory Finally, equation we consider the case N = ps, and study a degenerate Kirchhoff involving Trudinger-Moser nonlinearity In our best knowledge, it is the first time our problems are studied in this area CONTACT Nguyen Van Thin thinmath@gmail.com, thinnv@tnue.edu.vn © 2021 Informa UK Limited, trading as Taylor & Francis Group COMMUNICATED BY S D’Asero KEYWORDS Integro-differential operators; fractional p-Laplace; Fountain Theorem; Mountain Pass Theorem; Morse theory; Trudinger-Moser nonlinearity AMS SUBJECT CLASSIFICATIONS Primary: 35J60; 35R11; 35S15 W CHEN AND N VAN THIN On the p1 & &pm fractional Laplacian with critical Sobolev-Hardy exponent 1.1 Introduction and main results In this section, we first consider the existence of solutions for the Kirchhoff type equation involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev-Hardy exponent ⎧ |u(x) − u(y)|p1 ⎪ ⎪ M dx dy (− )sp11 u + · · · ⎪ ⎪ R2N |x − y|N+p1 s1 ⎪ ⎪ ⎪ |u(x) − u(y)|pm ⎪ ⎨ dx dy (− )spmm u +Mm N+pm sm 2N |x − y| (1) R ⎪ p∗s (α)−2 u q−2 u ⎪ |u| |u| ⎪ ⎪ = f (x, u) + γ +β in , ⎪ ⎪ ⎪ |x|α |x|α ⎪ ⎩ u = in RN \ , where < sm < · · · < s1 = s < 1, < pm ≤ · · · ≤ p1 = p < Ns , m ≥ 1, γ and β are posp(N−α) itive parameters and p∗s (α) = N−sp ≤ p∗s (0) is called the critical Sobolev-Hardy exponent, < q < p, < α < ps is an open bounded subset of RN with smooth boundary and ∈ M1 , , Mm are continuous functions which belong to the class M of continuous Kirchhoff functions M : [0, +∞) → (0, +∞) Here (− )sr u = lim ε→0+ RN \Bε (x) |u(x) − u(y)|r−2 (u(x) − u(y)) dy |x − y|N+rs with r ∈ {p1 , , pm } is the fractional r-Laplace, please see [1,2] and the references therein for more details Note that when s → 1− , m = and M = 1, (1) reduces to the p&q elliptic problem of the form − pu − u=0 qu = g(x, u) in , in ∂ The study of the above equation is motivated by the more general reaction-diffusion system which finds applications in biophysics, plasma physics and chemical reaction design, see [3] for more motivations In the last years, a great attention has been devoted to the study of classical p&q Laplacian problems in bounded domains and in RN ; see for instance [4–8] and the references therein Recently, Zhang [9] studied the existence and multiplicity of positive solutions for the fractional Laplacian problems with Hardy potentials and critical Sobolev-Hardy exponent by the Nehari manifold method and analytic techniques Ambrosio and Isernia [10] investigated the existence of solutions concerning the fractional p&q Laplacian problem with critical Hardy-Sobolev exponents by using the concentrationcompactness principle and the symmetric mountain pass lemma due to Kajikiya [11] In detail, they used a truncated functional and genus method to get the existence of infinitely many solutions which converge to zero For more related results, we refer the reader to [12–15] From our observation, there are no results for the existence of many solutions of the p&q fractional Laplacian with energy tending to infinitely COMPLEX VARIABLES AND ELLIPTIC EQUATIONS In the last years, great attention has been focused on the study of Kirchhoff type equations involving fractional and nonlocal operators This type of the problems arises in the description of nonlinear vibrations of an elastic string They have many different applications such as continuum mechanics, phase transition phenomena, population dynamics and game theory, as they are outcome of stochastically stabilization of Lévy processes [16–18] The literature on nonlocal operators and their applications is very interesting and quite large, we refer the interested reader to [19–22] and the references therein Furthermore, much interest towards p-Kirchhof type problems has grown more and more, we refer the reader to [23–25] and references therein Motivated by the above results, in this paper we are interested in the existence of solutions for Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev-Hardy exponent To our knowledge, no results concern Kirchhoff type equations involving nonlocal p1 & &pm fractional Laplacian with critical SobolevHardy exponent The main difficulty in our problems is due to the lack of compactness caused by the presence of the critical Hardy-Sobolev exponent and the fact that the nonlinear function does not satisfy the Ambrosetti-Rabinowitz condition We consider the class M of the Kirchhoff functions M which satisfy: (m1 ) M ∈ C(R+ ) satisfies inf t∈R+ M(t) ≥ m0 > 0, where m0 > is a constant; p (m2 ) There exists θ ∈ ( p∗ (α) , 1] such that M(t) = s t M(τ ) dτ ≥ θ M(t)t for all t ∈ R+ We assume that f is a Carathéodory function which satisfies the conditions: (f1 ) f : × R → R is a Carathéodory function and there exist q, with p < q < p∗s , a > Np such that |f (x, t)| ≤ a(1 + |t|q−1 ) for all x ∈ and for all t ∈ R, where p∗s = N−ps is the critical exponent of the fractional Sobolev space W s,p (RN ) p (f2 ) There exist μ : θ < μ < p∗s (α), and r > such that f (x, t)t − μF(x, t) ≥ −ρ|t|d − ϕ(x), for all x ∈ and |t| ≥ r, τ where F(x, t) = f (x, τ ) dτ , ρ ≥ 0, d ∈ [0, p] and ϕ(x) ≥ 0, ϕ ∈ L1 ( ) (f3 ) There exists δ > : |f (x, t)| ≤ pa|t|p−1 for all |t| ≤ δ and all x ∈ , where a > (f4 ) There exists γ∗ > such that F(x, t) ≥ γ∗ |t|μ − P(|t|) for all (x, t) ∈ × R, where P is a polynomial with degree ≤ deg P < μ Obviously, we can see that the condition (f2 ) is an extension of the Ambrosettiμ Rabinowitz condition The function [26] f (x, t) = sin |t| − |t| cos |t| + |t|μ , d = p = 2, p μ > θ satisfies the conditions (f1 )–(f4 ) and it does not satisfy the Ambrosetti-Rabinowitz condition Definition 1.1: We say that u ∈ X is a weak solution of problem (1) if m Mi (||u|| i=1 pi ) s ,p W0i i ( ) R2N |u(x) − u(y)|pi −2 (u(x) − u(y))(ϕ(x) − ϕ(y)) dx dy |x − y|N+pi si W CHEN AND N VAN THIN ∗ = |u|ps (α)−2 uϕ dx + β |x|α f (x, u(x))ϕ(x) dx + γ |u|q−2 uϕ dx |x|α for any ϕ ∈ X, where the space X will be introduced in Section 1.2 From Lemma 1.1 below in Section 2, we see, that the embedding X → Lν ( ) is continuous for all ν ∈ [1, p∗s ] Thus, we can define the best Sobolev constant: Sν = inf v∈X,v=0 |v(x)−v(y)|p RN ×RN |x−y|N+ps dx dy ( |v(x)|ν dx)p/ν (2) For all ≤ α ≤ ps, ≤ ν ≤ p∗s (α), we can see the embedding X → Lν ( , |x|−α ) is also continuous by Lemma 1.2 below in Section 2, thus the best Hardy constants are well defined: Aν,α = inf v∈X,v=0 |v(x)−v(y)|p RN ×RN |x−y|N+ps dx dy ν p/ν ( |v(x)| |x|α dx) (3) Before stating our first result, we need the assumption m0 θ − aS−1 p > p Theorem 1.2: Suppose that M1 , , Mm belong to the class M and f satisfies (f1 ), (f2 ) with d ∈ [0, p), (f3 ) and (f4 ), then there exists γ ∗ > such that problem (1) has a nontrivial weak solution in X for all γ ∈ (0, γ ∗ ) and β > small enough Next we consider the multiplicity solutions for problem (1) in the case γ = Namely, we consider the problem of ⎧ |u(x) − u(y)|p1 ⎪ ⎪ dx dy (− )sp11 u + · · · M ⎪ ⎪ R2N |x − y|N+p1 s1 ⎪ ⎪ ⎪ |u(x) − u(y)|pm ⎪ ⎨ dx dy (− )spmm u +Mm N+pm sm 2N |x − y| (4) R ⎪ ⎪ |u|q−2 u ⎪ ⎪ = f (x, u) + β in , ⎪ ⎪ ⎪ |x|α ⎪ ⎩ u = in RN \ We impose some more assumptions on f (f1 ) There exist constants < γ1 < γ2 < · · · < γm∗ < p < α∗ < p∗s , and functions α∗ h0 (x) ∈ Lβ∗ ( ), hi (x) ∈ L α∗ −γi ( ), i = 1, , m∗ , where stant hm∗ +1 > such that m |f (x, u)| ≤ |h0 (x)| + i=1 for all (x, u) ∈ × R α∗ + β∗ = 1, and a con- |hi (x)||u|γi −1 + hm∗ +1 |u|α∗ −1 , (5) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS (f5 ) f (x, −t) = −f (x, t) for x ∈ and t ∈ R Remark 1.3: The class of nonlinear functions which satisfy the condition (f1 ) contains many concave quantities Assume that ρ satisfies the condition: ⎧ ⎪ if d ∈ [0, p), ⎨ρ ∈ [0, ∞), −1 ρS θ p ⎪ − − > 0, if d = p ⎩m0 p μ μ (6) Theorem 1.4: Let M1 , , Mm belong to the class M Suppose that f satisfies (f1 ) , (f2 ) with p ρ satisfying (6) and θ < μ < p∗s , (f4 ) and (f5 ), then problem (4) has an infinite sequence of nontrivial weak solutions {uk } in Z( ) such that limk→∞ J(uk ) = +∞ In Theorem 1.4, Z( ) is a subspace of X which will be introduced in Section Recently, by using Nehari manifold and degree theory, Chen et al [3] proved the existence of positive and sign-changing solutions for the nonlinear nonlocal problems involving the fractional p-Laplacian with Sobolev and Hardy nonlinearity at subcritical and critical growth: ⎧ q−2 ⎨(− )s u = λ|u|r−2 u + μ |u| u in , p |x|α ⎩ u=0 in RN \ In this paper, using Morse theory which is very different from the methods used in [3], we study the existence of two nontrivial solutions of the problem ⎧ q−2 ⎨(− )s u = λf (x, u) + β |u| u in , p (7) |x|α ⎩ u=0 in RN \ , where λ and β are positive parameters In order to prove our result, we need some assumptions: (f6 ) There exists < η < such that F(x, t) ≥ a1 |t|p for all (x, t) ∈ × [−η, η], where a1 > p (f7 ) |f (x, t)| ≤ a(x)|t|p−1 for all (x, t) ∈ × R, where p ∈ (1, p) and a > 0, a ∈ L p−p ( ), and f (x, 0) = Theorem 1.5: Suppose that f : × R → R satisfy the conditions (f6 ) and (f7 ) Then the problem (7) has two nontrivial solutions in Z( ) for each positive parameters λ and β This paper is organized as follows In Section 1.2, we give some necessary definitions and properties of the space X In Section 1.3, by using Mountain Pass Theorem, we obtain the existence of solutions for the above problem In Section 1.4, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem as γ = In Section 1.5, we study the existence of two nontrivial solutions for Kirchhoff type equations involving nonlocal p-Laplacian via Morse theory W CHEN AND N VAN THIN 1.2 Some preliminary results s,p We denote X by the space W0 ( ) = {u ∈ W s,p (RN ) : u = in RN \ norm |u(x) − u(y)|p p u p = [u]s,p = dx dy N+ps R2N |x − y| } endowed with We also consider the fractional Sobolev space Z( ) as the closure of C0∞ ( ) with respect s,p s,p to above norm Then we have Z( ) ⊂ W0 ( ), with possibly Z( ) = W0 ( ), for more s ,p s ,q details, please see [14] By Lemma 2.2 in [27], we have W01 ( ) ⊂ W02 ( ) for all < q ≤ p, < s2 < s1 = s < From the assumption pm ≤ · · · ≤ p1 = p, then we get s ,p s ,p W01 ( ) ⊆ · · · ⊆ W0m m ( ) Therefore, we can study the weak solution of problem (1) in X We see that (X, X ) is a uniformly convex Banach space Set Q = R2N \ O, where O = C × C ⊂ R2N Then the norm on X can be written as follows u p p = [u]s,p = Q |u(x) − u(y)|p dx dy |x − y|N+ps Since × is strictly contained in Q, then from Corollary 7.2 in [28], we get the characteristic for embedding from X into Lν ( ), ν ∈ [1, p∗s ] as follows: Lemma 1.1: The following assertions hold true: (a) the embedding X → Lν ( ) is continuous for any ν ∈ [1, p∗s ]; (b) the embedding X → Lν ( ) is compact for all ν ∈ [1, p∗s ) Combining Theorem 2.5 [29], Lemmas 2.1 and 2.3 [3], we have the result as follows: Lemma 1.2: Two statements as follows hold: (a) For any ≤ α ≤ ps, X embeds continuously into Lq ( , |x|−α ) for all q ∈ (0, p∗s (α)] (b) Assume that < b < ps, and that ⊂ RN is an open bounded domain with smooth boundary and ∈ Then the embedding X → Lq ( , |x|−α ) is compact if ≤ q < q p∗s (b), ≤ α < sq + N(1 − p ) In fact, from [3], we just get that X embeds continuously into Lq ( , |x|−α ) for all q ∈ [1, p∗s (α)] In order to get our assertion a), we should use the inequality as follows |u|q dx = |x|α |u|q ∗ |x|αq/ps (α) ∗ ≤ · |u|ps (α) dx |x|α ∗ |x|α(1−q/ps (α)) q/p∗s (α) dx dx |x|α (p∗s (α)−q)/p∗s (α) q ≤ C[u]s,p holds for all < q ≤ p∗s (α), where C > is a suitable constant Remark 1.6: The result of Lemma 1.1 also comes from Lemma 1.2 when α = (8) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS The compactness assumption required by the Fountain Theorem is the well-known Palais-Smale condition (see, for instance, [30,31] and references therein), which in our framework reads as follows: Palais-Smale condition Let be a function in C1 (X, R) The function satisfies the Palais-Smale compactness condition at level c ∈ R if any sequence {uj }j∈N in X such that (uj ) → c and sup ϕ X =1 | (uj ), ϕ | → 0, admits a strongly convergent subsequence in X 1.3 Proof of Theorem 1.2 In order to study the existence of week solutions for problem (1), we consider the energy function on X as follows: m J(u) = i=1 − Mi pi R2N |u(x) − u(y)|pi dx dy |x − y|N+pi si F(x, u) dx − γ p∗s (α) ∗ |u|ps (α) β dx − |x|α q |u|q dx |x|α (9) Then from (f1 ), we have J ∈ C1 (X, R) Furthermore, we get J (u), ϕ m Mi ( u = i=1 pi ) s ,p W0i i ( ) R2N |u(x) − u(y)|pi −2 (u(x) − u(y))(ϕ(x) − ϕ(y)) dx dy |x − y|N+pi si ∗ − f (x, u)ϕ dx − γ |u|ps (α)−2 uϕ dx − β |x|α |u|q−2 uϕ dx |x|α for all u, ϕ ∈ X Certainly, the week solution of problem (1) is a critical point of the energy function J In order to prove Theorem 1.2, we need the Mountain Pass Theorem as follows: Lemma 1.3 ([30,31,51]): Let X be a real Banach space and I ∈ C1 (X, R) satisfies PalaisSmale condition (or (PS) condition) Suppose that I(0) = and (I1 ) There exist two constants β0 , α0 > such that I(u) ≥ α0 for all u = β0 (I2 ) There is u1 ∈ X : u1 ≥ β0 such that I(u1 ) ≤ Then I possesses a critical value c ≥ α0 Moreover, c can be characterized as c = inf max I(γ (t)), γ ∈ t∈[0,1] where = {γ ∈ C([0, 1], X) : γ (0) = 0, γ (1) = u1 } Lemma 1.4: Suppose that M1 , , Mm belong to the class M, f satisfies (f1 ) and (f3 ), and m0 θ −1 p − aSp > Then there is ρ > and c > such that J(u) ≥ c > for all u ∈ X with u X = ρ W CHEN AND N VAN THIN Proof: From the conditions (f1 ) and (f3 ), there exists C > such that |f (x, t)| ≤ pa|t|p−1 + qC|t|q−1 for all (x, t) ∈ (10) × R It follows from (10) that |F(x, t)| ≤ a|t|p + C|t|q (11) From the assumption of class M, we have Mi (t) ≥ m0 θ t for all i = 1, , m From (11), we deduce m J(u) ≥ m0 θ i=1 u pi u W0 ( ) for all ν s,p J(u) ≥ −1/p − u s,p W0 ( ) (12) for all ν ∈ [1, p∗s ] and s,p ∈ [1, p∗s (α)] Note that we denote X = W0 ( ), m0 θ p −q/p q − aS−1 u X − CSp u X p p γ β −q/p −p∗s (α)/p p∗ (α) Ap∗ (α),α − ∗ u Xs − Aq,α u s ps (α) q Since ≤ q < p, from (13) and p < p∗s (α) ≤ p∗s = J(u) ≥ u |u|q dx |x|α ≤ Sν Lp ( ) |u|q dx |u|p dx − C ∗ From (2) and (83), we have −1/p −a |u|ps (α) β dx − |x|α q γ − ∗ ps (α) u Lp ( ,|x|−α ) ≤ Aν,α u then from (12), we obtain pi s ,p W0i i ( ) m0 θ − aS−1 p p q X γ −p∗ (α)/p p∗s (α) s Ap∗ (α),α s u u Np N−ps p−q X p∗s (α)−q X − q X (13) (since ≤ α < ps), we get −q/p − CSp β −q/p Aq,α q q−q X (14) p∗s (α)−p on [0, +∞) We X m0 θ see that g is a continuous function on [0, +∞) and limt→0+ g(t) = p − aS−1 p > By m0 θ the continuity of g, for any ε > 0, there exists t0 > such that g(t) > p − aS−1 p − ε for m0 θ /p−aS−1 p all ≤ t ≤ t0 Specially, if we choose ε = , we have −q/p Set g(t) = ( mp0 θ − aS−1 p ) − CSp u g(t0 ) ≥ q−p X − m0 θ p −p∗s (α)/p γ p∗s (α) Ap∗s (α),α u − aS−1 p > If we choose β small enough such that q−p t0 m0 θ p − aS−1 p − β −q/p Aq,α > 0, q u COMPLEX VARIABLES AND ELLIPTIC EQUATIONS then from (14), we have q p−q J(u) ≥ t0 t0 when u X m0 θ p − aS−1 p − β −q/p Aq,α q =c>0 = ρ = t0 Lemma 1.5: Suppose that M1 , , Mm belong to the class M, and f satisfies (f4 ) Then there is e ∈ C0∞ ( ) such that e X ≥ ρ > and J(e) ≤ c, where ρ and c are give in Lemma 1.4 Proof: From the condition (m2 ), we have Mi (t) ≤ Mi (1)t 1/θ for any t ≥ and i = 1, , m We can fix u0 ∈ C0∞ ( ) such that u0 X = By Lemma 2.2 in [27], there exits a constant C ≥ such that u s ,p W0i i ( ) ≤C u (15) X for all i = 1, , m For all t ≥ 1, from condition (f4 ) and (15), we have m J(tu0 ) = i=1 pi )− s ,p W0i i ( ) m ≤ i=1 F(x, tu0 ) dx ∗ |u|ps (α) β dx − t q |x|α q γ p∗s (α) t p∗s (α) − − M( tu0 pi C pi /θ Mi (1)tpi /θ − γ∗ tμ pi |u|q dx |x|α |u0 |μ dx + ∗ |u|ps (α) β dx − t q |x|α q γ p∗s (α) t p∗s (α) P(|tu0 |) dx |u|q dx |x|α Note that μ > p/θ and ≤ deg P < μ, we have J(tu0 ) → −∞ as t → +∞ Hence we may choose e = Tu0 , where T > is sufficiently large Lemma 1.6 ([10,33]): Let {un }n be a bounded sequence in X and α ∈ [0, ps] Then, up to a subsequence, there exists u ∈ X, two Borel regular measures η and ν, J denumerable, xj ∈ , νj ≥ 0, ηj ≥ with νj + ηj > 0, j ∈ J such that un → u weakly in X, RN |un (x) − un (y)|p dy |x − y|N+ps dη ≥ RN ∗ ∗ dη, |u(x) − u(y)|p dy + |x − y|N+ps |un |ps (α) |x|α ηj δxj , ∗ dν, (16) ηj := η({xj }), (17) j∈J ∗ dν = ηj ≥ |u|ps (α) + |x|α νj δxj , νj := ν({xj }), (18) j∈J p/p∗ (α) Ap∗s (α),α νj s (19) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 31 Trudinger-Moser nonlinearities In our knowledge, no results concern singular Kirchhofftype equations involving nonlocal fractional p-Laplacian with Trudinger-Moser nonlinearities on bounded domain Using Theorem 1.2, we study the existence the solution of problem (78) via Mountain Pass Theorem We not try improve the hypothesis due to Xiang-Radulescu-Zhang [48] In order to study the problem (78), we consider some assumptions on f as follows: (h1 ) The nonlinearity f : × R → R is continuous, and there exist constants α0 ∈ (0, β∗ ), b1 , b2 > such that for any (x, t) ∈ × R, |f (x, t)| ≤ b1 |t|θp−1 + b2 N,s (y) where = ey − (h2 ) There exist μ > θ Ns jp −2 yj i=0 j! , jp N,s (α0 |t| N/(N−s) ), = min{j ∈ N : j ≥ p} such that f (x, t)t ≥ μF(x, t) > for all (x, t) ∈ × R, t = 0, where F(x, t) = (h3 ) There exists a constant B > such that lim sup t→0 uniformly in x ∈ , where M(1)s N τ f (x, τ ) dτ F(x, t) 0, AθN/s,γ is defined by (83) s,p Definition 2.2: We say that u ∈ W0 ( ) is a weak solution of problem (78) if M( u p ) s,p W0 ( ) R2N |u(x) − u(y)|p−2 (u(x) − u(y))(ϕ(x) − ϕ(y)) dx dy |x − y|N+ps f (x, u(x)) ϕ(x) dx + β |x|γ = |u|q−2 uϕ dx |x|α s,p for any ϕ ∈ W0 ( ) Now, our result on equation (78) is given as follows: Theorem 2.3: Let (m3 ), (m4 ), (h1 )–(h3 ) hold Then there exists > such that the problem (78) admits a nontrivial solution in uβ ∈ X for all β ≥ Moreover limβ→+∞ uβ X = This part is organized as follows In Section 2.2, we give some preliminary results about embedding on X and we give the proof of Theorem 2.1 in Section 2.3 In Section 2.4, by using Mountain Pass Theorem, we obtain the existence of solution for the problem (78) 32 W CHEN AND N VAN THIN 2.2 Some preliminary results Lemma 2.1 ([48]): Let s ∈ (0, 1) and N ≥ Suppose that is bounded domain in RN s,N/s with Lipschitz boundary ∂ Then the embedding W0 ( ) → W s,N/s ( ) → → Lν ( ) is compact for all ν ∈ [1, +∞) s,N/s Lemma 2.2: Let q ≥ and ≤ α < N We have W0 ( ) embeds compactly into s,N/s Lq ( , |x|−α ) Hence W0 ( ) embeds compactly into Lq ( , |x|−α ) Proof: If α = 0, we have done by Lemma 2.1 Now, we consider N > α > First we show s,N/s that W0 ( ) embeds continuously into Lq ( , |x|−α ) It is enough to show that |u|q dx |x|α 1/q ≤C u s,N/s W0 ( ) , (80) where C > is a constant Choose t > such that tα < N, apply Hölder inequality t and t with 1t + t1 = 1, we have |u|q dx ≤ |x|α |u|qt dx 1/t dx |x|αt 1/t (81) s,N/s Note that tα < N, then dx is finite For any q ≥ 1, by Lemma 2.1, W0 ( ) is |x|αt continuously and compactly embedded into Lq ( ) Therefore, (81) implies that there exists s,N/s C > such that (80) holds It means that the embedding from W0 ( ) into Lq ( , |x|−α ) is continuous s,N/s If un converges weakly to u in W0 ( ), then un − u converges weakly to in s,N/s W0 ( ) Therefore, we only need prove Lemma 2.1 for the case of a sequence un → s,N/s weak in W0 ( ) We claim that |un |q dx → |x|α as n → ∞ (82) From (81), Lemma 2.1, and note that qt > q, we get lim n→∞ |un |q dx ≤ lim n→∞ |x|α |un |qt dx 1/t dx |x|αt 1/t = 0, which implies (82) For for all ≤ α < N, ≤ ν < +∞, we can see the embedding X → Lν ( , |x|−α ) is also continuous by Lemma 2.2, and the following best Hardy constant is well defined: Aν,α = inf v∈X ,v=0 When α = 0, we denote Aν,0 by Sν |v(x)−v(y)|p RN ×RN |x−y|N+ps dx dy p/ν |v(x)|ν dx α |x| (83) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 33 s,p Lemma 2.3: For any α > and any u ∈ W0 ( ), we have N/(N−s) ) N,s (α|u| |x|γ dx < +∞ Proof: We have N/(N−s) ) N,s (α|u| |x|γ eα|u| |x|γ N/(N−s) dx ≤ dx (84) It is enough to show that eα|u| |x|γ N/(N−s) dx < +∞ s,p for any α > Fix α > 0, u ∈ W0 ( ) and ε > 0, there exists ϕ ∈ C0∞ ( ) such that u − ϕ W s,p ( ) < ε We have eα|u| N/(N−s) = eα|u−ϕ+ϕ| ≤ N/(N−s) ≤ eα2 N/(N−s) |u−ϕ|N/(N−s) 2α2N/(N−s) |u−ϕ|N/(N−s) 2e + 12 e2α2 · eα2 N/(N−s) |ϕ|N/(N−s) N/(N−s) |ϕ|N/(N−s) It follows that eα|u| |x|γ N/(N−s) dx ≤ e2α2 N/(N−s) |u−ϕ|N/(N−s) |x|γ 2α2N/(N−s) u−ϕ = e + e2α2 dx + s,N/s W0 ( ) N/(N−s) |ϕ|N/(N−s) N/(N−s) ( u−ϕ |u−ϕ| )N/(N−s) s,N/s W0 ( ) |x|γ e2α2 e dx |x|γ (85) s,N/s ( ) W0 N/(N−s) ( u−ϕ |u−ϕ| )N/(N−s) s,N/s W0 ( ) |x|γ Using Hölder inequality for t > : tγ < N, t > : e2α2 N/(N−s) |ϕ|N/(N−s) |x|γ dx ≤ dx N/(N−s) |ϕ|N/(N−s) Choose ε > small enough such that 2α(2ε)N/(N−s) ≤ β∗ < α∗ (1 − Theorem 1.2, we have 2α2N/(N−s) u−ϕ dx |x|γ e2t α2 t + N/(N−s) |ϕ|N/(N−s)dx t dx < +∞ then by (86) = 1, we have 1/t From (84) to (87), we get N/(N−s) ) N,s (α|u| |x|γ γ N ), dx < +∞ dx |x|γ t 1/t < +∞ (87) 34 W CHEN AND N VAN THIN 2.3 Proof of Theorem 2.1 First, if γ = 0, we get the statement of Theorem 2.1 from Theorem D and [42] We consider the case γ ∈ (0, N) For all α < α∗ (1 − Nγ ), choose t > such that Nγ < 1t < − αα∗ , apply Hölder inequality t and t with 1t + t1 = 1, we have exp(α|u|N/(N−s) ) dx ≤ |x|γ exp α t |u|N/(N−s) t−1 (t−1)/t dx dx |x|γ t 1/t (88) Since be a bounded open domain of RN with Lipschitz boundary, then dx < |x|γ t t +∞ From the condition of t, we have α t−1 < α∗ , then by Theorem D and [42], we get sup s,p u∈W0 ( ),[u]W s,p (RN ) ≤1 exp(α|u|N/(N−s) ) dx < +∞ |x|γ ∗ , we have Next we show that for all α > αs,N N/(N−s) ) N,s (α|u| |x|γ sup s,p u∈W0 ( ),[u]W s,p (RN ) ≤1 Without loss of generality, we may assume that introduced in [32] by uε (x) = −N/(N−s) is the unit ball Denote the same sequence ⎧ | ln ε|(N−s)/N ⎪ ⎪ ⎨ | ln |x if |x| ≤ ε if ε ≤ |x| < ⎪ | ln ε|s/N ⎪ ⎩ From [32], we have [uε ]W s,p (RN ) → ( enough such that dx = +∞ if |x| > ∗ αs,N (N−s)/N N ) ∗ , we can choose ε small Since α > αs,N ≥ α|uε (x)|N/(N−s) e |uε | )N/(N−s) ) and N/(N−s) ) N,s (α|uε (x)| )N/(N−s) ) N,s (α( [uε ] α[uε ]W s,p (RN ) ≥ β > N Hence, we obtain N,s (α( [uε ] |uε | W s,p (RN ) |x|γ dx > ≥ |x|γ Bε (0) ≥ W s,p (RN ) dx N/(N−s) ) ε (x)| exp(α( [uε|u ] s,p N ) W Bε (0) exp(−β ln ε) (R ) |x|γ Bε (0) dx |x|γ dx (89) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS Since dx = N| | |x|γ Bε (0) ε N| | N−γ ε N−γ rN−γ −1 dr = 35 (90) Combine (89) and (90), from the condition ≤ γ < N and β > N, we get N,s (α( [uε ] |uε | W s,p (RN ) )N/(N−s) ) |x|γ dx ≥ N| | exp(−β ln ε)ε N−γ → +∞ 2(N − γ ) as ε → 0+ 2.4 Proof of Theorem 2.3 In order to study solution of problem (78) we consider the energy function Iβ (u) = M p R2N |u(x) − u(y)|p dx dy − |x − y|N+ps |u|q dx |x|α F(x, u) β dx − |x|γ q (91) From the condition (h3 ), there exist τ > and δ > such that for all |t| ≤ δ, we have |F(x, t)| ≤ (B − τ )|t|θN/s (92) for all x ∈ Moreover from the condition (h1 ) and f is a continuous function, for each q > θ Ns , we can find a constant C = C(q, δ) > such that |F(x, t)| ≤ C|t|q for all |t| ≥ δ and x ∈ N,s (α0 |t| N/(N−s) (93) Combine (92) and (93), we get |F(x, t)| ≤ (B − τ )|t|θN/s + C|t|q for all x ∈ and t ∈ R From (94) and Lemma 2.3, for t > and t > : we deduce |u|q ) N,s (α0 |u| |x|γ N/(N−s) ) ≤ = N/(N−s) ) t N,s (α0 |u| |x|γ t + |u|q |x|γ /t dx = N/(N−s) ) N,s (α0 |u| |x|γ /t N,s (α0 |t| 1/t t t N/(N−s) (94) = 1, using Hölder inequality, N/(N−s) ) N,s (α0 |u| |x|γ /t |u|qt dx |x|γ dx ) dx 1/t 1/t dx u q Lq t ( ,|x|−γ ) (95) By Lemma 2.3 [50], for any b > t, there exist a constant C(b) > such that ( N,s (α0 |u| N/(N−s) ))t ≤ C(b) s,p W0 ( N,s (bα0 |u| N/(N−s) ) (96) on RN Since the embedding from ) → Lqt ( , |x|−γ ) is continuous, combine Lemma 2.3, (95) and (96), there exists a constant D > such that |u|q N,s (α0 |u| |x|γ N/(N−s) ) dx 36 W CHEN AND N VAN THIN N,s (bα0 |u| |x|γ ≤ C(b) 1/t ≤D u q s,p W0 ( ) N/(N−s) ) 1/t dx u q Lq t ( ,|x|−γ ) < +∞ (97) s,p Hence, the energy function Iβ in (91) is well defined on W0 ( ) Compute similarly to s,p Lemmas 3.1 and 3.2 [41], we have Iβ ∈ C1 (W0 ( ), R) and Iβ (u), ϕ = M( u N/s ) s,p W0 ( ) R2N |u(x) − u(y)|p−2 (u(x) − u(y))(ϕ(x) − ϕ(y)) dx dy |x − y|N+ps f (x, u(x))ϕ(x) dx − β |x|γ − |u|q−2 uϕ dx |x|α s,p for all ϕ ∈ W0 ( ) Certainly, the week solution of problem (78) is a critical point of the energy function Iβ In order to prove Theorem 2.3, we need the following lemma Lemma 2.4: Suppose that (m1 ), (m2 ), (h1 ) and (h3 ) hold Then there exist constants positive t0 , ρ0 (β) such that Iβ (u) ≥ ρ0 for all u ∈ X , with u X = t0 Proof: From (94), we have |F(x, t)| ≤ (B − τ )|t|θN/s + C|t|q N/(N−s) ) and t ∈ R From the condition (m4 ), we have for all x ∈ M(t) ≥ M(1)tθ , Then for u X t ∈ [0, 1] is small, we get Iβ (u) = M( u p ≥ N,s (α0 |t| p )− s,p W0 ( ) M(1)s θN/s u s,p W0 ( N q |u| dx −β |x|α ) F(x, u) dx − β |x|γ |u|q dx |x|α |u|θN/s dx − C |x|γ − (B − τ ) |u|q N/(N−s) ) N,s (α0 |u| |x|γ dx (98) From (95), we have |u|q N/(N−s) ) N,s (α0 |u| |x|γ N/(N−s) ) N,s (α0 |u| |x|γ dx ≤ where t > : tγ < N, t > and a constant C(b) > such that t + t N/(N−s) ) N,s (α0 |u| |x|γ 1/t t dx u q , Lq t (R N ) (99) = By Lemma 2.3 [50], for any b > t, there exist t ≤ C(b) N/(N−s) ) N,s (bα0 |u| |x|γ t (100) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS on 37 Therefore, we get t N/(N−s) ) N,s (α0 |u| γ |x| N,s (bα0 = C(b) When u s,N/s W0 N/(N−s) ) N,s (bα0 |u| γ t |x| ≤ C(b) u N/(N−s) |u/ s,N/s W0 ( ) |x|γ t u s,N/s W0 ( ) dx |N/(N−s) ) dx (101) is small enough and b near t, we have ( ) bα0 u N/(N−s) dx s,N/s W0 ( ) γ , N ≤ β∗ < α∗ − (102) by Theorem 2.1, there exits a constant D > such that s,N/s Since the embedding from W0 |u|q N,s (α0 |u| |x|γ 1/t t N/(N−s) ) N,s (α0 |u| |x|γ ≤ D dx ( ) → Lqt ( ) is continuous, we get N/(N−s) ) −q dx ≤ DSqt u q s,N/s W0 ( ) < +∞ (103) ( ) = X (104) From (83), we have u LθN/s ( ,|x|−ν ) ≤ A−1 θN/s,ν u s,N/s for all u ∈ W0 X Hence, combine (98), (103) and (104), we obtain Iβ (u) ≥ M(1)s u N = u θN/s X θN/s X −θ N/s − (B − τ )AθN/s,γ u θN/s X −q − CDSqt u M(1)s −q −θ N/s − (B − τ )AθN/s,γ − CDSqt u N q X q−θ Ns X − β −q Aq,α u q − β −q Aq,α u q q X q−θ Ns X (105) Since M(1)s N −θ N/s − BAθN/s,γ > 0, then we have M(1)s −θ N/s − (B − τ )AθN/s,γ > N Let h(t) = M(1)s N −θ N/s −q − (B − τ )AθN/s,γ − CDSqt t q−θ N s −q − βq Aq,α t q−θ s , t ≥ We now prove N −θ N/s there exists t0 > small satisfying h(t0 ) ≥ 12 ( MN(1)s − (B − τ )AθN/s,γ ) We see that h is continuous function on [0, +∞) and limt→0+ h(t) = β > 0, then there exists t0 (β) such that h(t) ≥ M(1)s N M(1)s N −θ N/s − (B − τ )AθN/s,γ for each −θ N/s − (B − τ )AθN/s,γ − ε1 for all ≤ 38 W CHEN AND N VAN THIN t ≤ t0 , t0 is small enough such that u −θ N/s (B − τ )AθN/s,γ ), we have h(t) ≥ X = t0 satisfies (102) If we choose ε1 = 12 ( MN(1)s − M(1)s −θ N/s − (B − τ )AθN/s,γ N M(1)s −θ N/s − (B − τ )AθN/s,γ N for all ≤ t ≤ t0 Especialy, h(t0 ) ≥ From (105) and (106), for u = t0 , we have X N/s Iβ (u) ≥ (106) t0 · M(1)s −θ N/s − (B − τ )AθN/s,γ N = ρ0 Lemma 2.5: Suppose that M satisfies (m3 ) and (m4 ), f satisfies (h1 ) − (h2 ) Then there is e ∈ C0∞ ( ) such that e X ≥ ρ0 > and Iβ (e) < where ρ is given in Lemma 2.4 Proof: It follows from the condition (m4 ) that M(t) ≤ M(1)t θ , t ≥ From the conditions (h1 ) − (h2 ), as in [41], there exists C1 > and C2 > such that F(x, t) ≥ C1 |t|μ − C2 Choose u ∈ C0∞ ( ) with u Iβ (tu) = ≤ X for all x ∈ and t ∈ R = 1, from for all t ≥ 1, we get s M( tu N F(x, tu) β dx − γ |x| q N/s X )− M(1)s θN/s t − C1 t μ N |tu|q dx |x|α |u(x)|μ dx + C2 | | |x|γ Since μ > θ Ns , we have Iβ (tu) → −∞ as t → +∞ Taking e = ρ1 u, ρ1 > ρ0 > large enough, we have Iβ (e) < 0, e X > ρ0 By Lemma 1.3, there exists a (PS) sequence {un } ⊂ X such that Iβ (un ) → cβ and Iβ (un ) → as n → ∞, where cβ = inf max Iβ (η(t)), η∈ t∈[0,1] (107) and = {η ∈ C([0, 1]; X ) : η(0) = 1, η(1) = e} Obviously, cβ > by Lemma 2.4 Further, we have the result as follows: Lemma 2.6: Suppose that (m4 ) and (h2 ) hold Then limβ→+∞ cβ = 0, where cβ is given by (107) COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 39 Proof: For e given by Lemma 2.5, we have limt→+∞ Iβ (te) = −∞ Therefore, by Lemma 2.4, there exists tβ > such that Iβ (tβ e) = maxt≥0 Iβ (te) Hence Iβ (tβ e) = 0, which implies that N/s tβ M( tβ e N/s X ) e N/s f (x, tβ e) q tβ e dx + βtβ γ |x| = |e|q dx |x|α (108) Let us first prove {tβ }β is bounded sequence Conversely, we assume that there exists a subsequence of {tβ }β , still denote by {tβ }β such that tβ → +∞ as β → +∞ By the condition (m4 ) and (108), we get θN/s θ M(1)tβ e θN/s X q ≥ βtβ |e|q dx, |x|α (109) thanks to the condition (h2 ) Note that q > θ N/s, we get tβ → as β → +∞, which is a contradiction Therefore, {tβ }β must be bounded From (109), we also have limβ→+∞ tβ = Set η∗ (t) = te Clearly, η∗ ∈ , thanks to the continuity of M, we have < cβ ≤ max Iβ (η∗ (t)) = Iβ (tβ e) ≤ M( tβ e t≥0 p N/s X ) →0 as β → +∞ This lemma is proved Lemma 2.7: Let {un }n ⊂ X be a (PS)cβ of Iβ Then there exists β > , up to a subsequence still denoted by {un }n , we have lim sup un n→∞ X < β∗ cα0 > such that for all (N−s)/N for any fixed c > Proof: First, we assume that d = inf n≥1 un X > We prove that {un }n is a bounded sequence Since {un }n is (PS)cβ sequence, then we have Iβ (un ) − I (un ), un ≤ cβ + o(1) + o(1) un μ β X Now, we consider two cases of μ First θ N/s < μ ≤ q, from the condition (h2 ), we have Iβ (un ) − s − M( un θN μ I (un ), un ≥ μ β + N/s X ) un N/s X f (x, un )un − μF(x, un ) dx + μ ≥ s − M( un θN μ ≥ μs − θ N μθ N N/s X ) κ(dN/s ) un which means that {un }n is bounded sequence in X un N/s X , 1 − β μ q |un |q dx |x|α N/s X (110) 40 W CHEN AND N VAN THIN If μ > q, then we have Iβ (un ) − s − M( un θN q I (un ), un ≥ q β N/s X ) un N/s X (111) f (x, un )un − qF(x, un ) dx q + ≥ s − M( un θN q ≥ qs − θ N qθ N N/s X ) κ(dN/s ) un un N/s X N/s X , (112) which means that {un }n is bounded sequence in X Since limβ→+∞ cβ = 0, then there exists > such that (110) and (112) imply lim sup un n→∞ X < ε(β) < β∗ cα0 (N−s)/N for any fixed c > and for all β ≥ If d = inf n≥1 un X = 0, then there exists a subsequence of {un }n , still denoted by {un }n such that limn→∞ un X = or d = inf n≥1 un X > The second case later is solved as before We have finished the proof of Lemma 2.7 Lemma 2.8: Suppose that (h1 )–(h2 ) hold Then the functional Iβ satisfies the (PS)cβ condition when β > Proof: Let {un }n be a (PS)cβ sequence Then by Lemma 2.7, passing to a subsequence if necessary, we can assume that un X → d ≥ If d = 0, then un → 0, which implies limn→∞ Iβ (un ) = = cβ , which is a contradiction Thus, we will consider the case d > Since X is separated and uniformly convex space, then up to a subsequence, thanks to Lemma 2.2, for any ν ≥ 1, we have un → u weak in X , un → u in Lν ( , |x|−γ ) un → u in We denote Bϕ the linear functional on X as follows N Bϕ (v) = R2N |ϕ(x) − ϕ(y)| s −2 (ϕ(x) − ϕ(y))(v(x) − v(y)) dx dy |x − y|2N Clearly, by Hölder inequality, Bϕ is a continuous linearly mapping on X and |Bϕ (v)| ≤ ϕ N s −1 X v X for all v ∈ X From the condition (h1 ) and using Hölder inequality, we get |(f (x, un ) − f (x, u))(un − u)| dx |x|γ COMPLEX VARIABLES AND ELLIPTIC EQUATIONS b1 |un |θp−1 + b2 ≤ b1 |u|θp−1 + b2 + ≤ b1 ( un θp−1 Lθp ( ,|x|−γ ) N/(N−s) ) θp−1 ) Lθp ( ,|x|−γ ) N/(N−s) ) + |un − u| dx |un − u| dx un − u N,s (α0 |u| Lθ p ( ,|x|−γ ) N/(N−s) ) |x|γ Using Hölder inequality for q ≥ N,s (α0 |un | |x|γ = N/(N−s) ) N,s (α0 |u| |x|γ + u N,s (α0 |un | + b2 ≤ N,s (α0 |un | |x|γ N/(N−s) ) N s = p, note that N/(N−s) ))q N,s (α0 |un | |x|γ ( N/(N−s) ))q N,s (α0 |un | |x|γ + q |un − u| dx 1/q dx 1/q |un − u|q dx |x|γ dx (113) = 1, we deduce N/(N−s) ) |u − u| N,s (α0 |un | n |x|γ /q |x|γ /q |un − u| dx = ( q 41 1/q dx un − u Lq ( ,|x|−γ ) (114) By Lemma 2.3 [50], choose c > q > 1, c is near q , there exist a constant C(c) > such that ( N,s (α0 |u| N/(N−s) ))q ≤ C(c) N,s (cα0 |u| N/(N−s) ) (115) for all u ∈ X By Lemma 2.7, we have c · α0 sup un n N/(N−s) X < β∗ (116) Then, apply to Theorem 2.1, we deduce N,s (cα0 |un | |x|γ sup n = sup N/(N−s) ) N,s (cα0 un n dx N/(N−s) (u/ X γ |x| un N/(N−s) ) X) dx < +∞ From (113) and (117), there exists a constant D1 > such that |(f (x, un ) − f (x, u))(un − u)| dx |x|γ ≤ b1 ( un θp−1 Lθp ( ,|x|−γ ) + D1 un − u + u Lq ( ,|x|−γ ) θp−1 ) Lθp ( ,|x|−γ ) un − u Lθ p ( ,|x|−γ ) (117) 42 W CHEN AND N VAN THIN Since the embedding X into Lν ( , |x|−γ ) is compact for any ν ∈ [1, +∞) and { un bounded sequence, we get X } is (f (x, un ) − f (x, u))(un − u) dx = |x|γ lim n→∞ Furthermore, the embedding from X → Lq ( , |x|−α ) is compact, using Brezis-Lieb Lemma, we obtain (|un |q−2 un − |u|q−2 u)(un − u) dx = |x|α lim n→∞ Obviously, Iβ (un ) − Iβ (u), un − u → since un → u weak in X and Iβ (un ) → Therefore, we get o(1) = Iβ (un ) − Iβ (u), un − u = M( un − p X )Bun (un − u) − M( u p X )Bu (un (f (x, un ) − f (x, u))(un − u) dx − |x|γ − u) (|un |q−2 un − |u|q−2 u)(un − u) dx |x|α p X )(Bun (un − u) − Bu (un − u)) p p + (M( un X ) − M( u X ))Bu (un − u) + o(1) p = M( un X )(Bun (un − u) − Bu (un − u)) + o(1) = M( un Since M( un p X) → M(dN/s ) > 0, then lim Bun (un − u) − Bu (un − u) = n→∞ It is well-know that the Simion inequalities |ξ − ν|l ≤ cl (|ξ |l−2 ξ − |ν|l−2 ν)(ξ − ν), for l ≥ 2, |ξ − ν|l ≤ Cl [(|ξ |l−2 ξ − |ν|l−2 ν)(ξ − ν)]l/2 (|ξ |l + |ν|l ) 2−l , for < l < 2, for all ξ , ν ∈ RN , where cl , Cl are positive constants depending only on l Using Simion inequality and by arguments as [41,47], we obtain un → u strong in X Now, we prove Theorem 2.3, by Lemmas 2.4 and 2.5, there exists (PS)cβ sequence of Iβ From Lemma 2.7, Iβ satisfies the (PS) condition with level cβ for all β ≥ Therefore, Iβ admits a nontrivial critical point uβ ∈ X by Lemma 1.3, which implies that Iβ (uβ ) = cβ and Iβ (uβ ) = in X ∗ Next, by argument as Lemma 2.6 for two cases θN/s < μ ≤ q and μ > q, we get limβ→+∞ uβ X = Acknowledgements The authors wish to thank the editorial board and referees for a very careful reading of the manuscript, and for pointing out misprints that led to the improvement of the original manuscript COMPLEX VARIABLES AND ELLIPTIC EQUATIONS 43 Disclosure statement No potential conflict of interest was reported by the author(s) Funding Nguyen Van Thin is supported by Ministry of Education and Training of Vietnam under project with the name ‘Weak solutions to some class equations, system of partial differential equations containing fractional p-Laplace and Bessel operator’ and [grant number B2020-TNA-06] Wei Chen is supported by the Science and Technology Research Program of Chongqing Municipal Education Commission [grant number KJQN202000621], the Fundamental Research Funds of Chongqing University of Posts and Telecommunications [grant number CQUPT:A2018-125] and the Basic and Advanced Research Project of CQCSTC [grant number cstc2019jcyj-msxmX] References [1] Iannizzotto A, Liu S-B, Perera K Existence results for fractional p-Laplacian problems via Morse theory Adv Calc Var 2016;9:101–125 [2] Lindgren E, Lindqvist P Fractional eigenvalues Calc Var Partial Differ Equ 2014;49:795–826 [3] Chen W, Mosconi S, Squassina M Nonlocal problems with critical Hardy nonlinearity J Funct Anal 2018;275(11):3065–3114 [4] Barile S, 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Mathematical Society; 1986 (CBMS regional conference series in mathematics; 65) ... in the existence of solutions for Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev- Hardy exponent To our knowledge, no results concern Kirchhoff. .. of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev- Hardy exponent Received 20 January... ELLIPTIC EQUATIONS https://doi.org/10.1080/17476933.2021.1913129 Existence of solutions to Kirchhoff type equations involving the nonlocal p1 & &pm fractional Laplacian with critical Sobolev- Hardy