FRACTIONAL p -SUB-LAPLACIAN OPERATOR PROBLEM WITH CONCAVE-CONVEX NONLINEARITIES ON HOMOGENEOUS GROUPS

. This study examines the existence and multiplicity of non-negative solutions of the following fractional p -sub-Laplacian problem where Ω is an open bounded in homogeneous Lie group G with smooth boundary, p > 1, s ∈ (0 , 1), ( − ∆ p,g ) s is the fractional p -sub-Laplacian operator with respect to the quasi-norm g , λ > 0, 1 < α < p < β < p ∗ s , p ∗ s := QpQ − sp is the fractional critical Sobolev exponents, Q is the homogeneous dimensions of the homogeneous Lie group G with Q > sp , and f , h are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter λ belong to a center subset of (0 , + ∞ ).

1. Introduction. We consider the following p-fractional Laplace equation where Ω is an open bounded domain in homogeneous Lie group G with smooth boundary, p > 1, the parameter λ > 0, f and h are sign-changing smooth functions, 1 < α < p < β < p * s := Qp Q−ps , p * s is the fractional critical Sobolev exponent in this context and Q > sp is the homogeneous dimension of the homogeneous Lie group G. The operator (−∆ p,g ) s is the fractional p-sub-Laplacian operator on G which is defined by where B g (x, ε) is the quasi-ball of center x ∈ G and radius ε > 0 with respect to the homogeneous quasi-norm g. In our work, the homogeneous quasi-norm g : G → R + (i) g(x) = 0 if and only if x = 0 for every x ∈ G; (ii) g(x −1 ) = g(x) for every x ∈ G; (iii) g(δ µ (x)) = µg(x) for every µ > 0 and for every x ∈ G, where δ µ is a dilations on homogeneous Lie group G. Associated with (1.1), we have the energy functional I λ : E 0 g → R defined by By a direct calculation, we have that I λ ∈ C 1 (E 0 g , R) and where E 0 g is a subspace of E g defined as E 0 g = {u ∈ E g : u = 0 in G\Ω} with the norm Here Q = G 2 \(C Ω × C Ω ) and C Ω = G\Ω. See Section 2 for more details.
Recently, a lot of attention is given to the study of fractional operators of elliptic type due to concrete real world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc. Dirichlet boundary value problem in case of fractional Laplacian with polynomial type nonlinearity using variational methods is recently studied in [4,6,21,20,19]. For example, Brändle et. al [4] studied the fractional Laplacian operator (−∆) s equation involving concave-convex nonlinearity for the subcritical case in the Euclidean space R N , they prove that there exists a finite parameter Λ > 0 such that for each λ ∈ (0, Λ) there exist at least two solutions, for λ = Λ there exists at least one solution and for λ ∈ (Λ, +∞) there is no solution. Barrios et al. [2] studied the non-homogeneous equation involving fractional Laplacian and proved the existence and multiplicity of solutions under suitable conditions of s and q. Zhang, Liu and Jiao [23] studied the fractional equation with critical Sobolev exponent, they proved that the existence and multiplicity of solutions under appropriate conditions on the size of λ. For more other advances on this topic, see [19] for the subcritical, [20] for the critical case, [22] for the supercritical case, and fractional Laplacian equation with Hardy-type potential are shown in [13,14,24,25,26,27]. Moreover, for the fractional p-Laplacian equation, eigenvalue problem related to p-fractional Laplacian is studied in [17,12]. Goyal and Sreenadh [15] studied the fractional p-Laplacian equation involving concave-convex nonlinearities. By using the Nehari manifold and the fibering maps methods, they showed that the problem has at least two non-negative solutions.
In this paper, we present results concerning fractional forms Laplacian operator on homogeneous Lie groups. As usual, the general approach based on homogeneous Lie groups allows one to get insights also in the Abelian case, for example, from the point of view of the possibility of choosing an arbitrary quasi-norm. Moreover, another application of the setting of homogeneous Lie groups is that the results can be equally applied to elliptic and subelliptic problems. We start by discussing fractional Sobolev inequalities on the homogeneous Lie groups. As a consequence of these inequalities, we derive the existence results for the nonlinear problem with fractional p-sub-Laplacian operator and concave-convex nonlinearities and sign-changing weight functions. We also extend this analysis to equations of fractional p-sub-Laplacians and to Riesz type potential operators.
To the best of our knowledge there is no work for fractional p-sub-Laplacian operator with convex-concave type nonlinearity and sign changing weight functions on the homogeneous Lie groups. We have the following existence result. Theorem 1.1. Let G be a homogeneous Lie group with homogeneous dimension Q, and let s ∈ (0, 1), Q > sp, 1 < α < p < β < p * s and f ∈ L Then there exists Λ * > 0 such that the equation (1.1) admits at least two non-negative solutions for λ ∈ (0, Λ * ).
The paper is organized as follows: In Section 2, we study the properties of the Sobolev spaces W s,p g (G) and E 0 g on homogeneous groups. In Section 3, we introduce Nehari manifold and study the behavior of Nehari manifold by carefully analyzing the associated fibering maps on homogeneous Lie groups. Section 4 contains the existence of nontrivial solutions in N + λ and N − λ .
2. Functional analytic settings on homogeneous Lie groups. In this section we discuss nilpotent Lie algebras and groups in the spirt of Folland and Stein's book [11] as well as introducing homogeneous Lie groups. For more analyses and details in this direction we refer to the recent open access book [10] and [1,5,3,7,8,9,18] and references therein. Let g be a real and finite-dimensional Lie algebra, and let G be the corresponding connected and simply-connected Lie group. The lower central series of g is defined inductively by g (1) := g, g (j) := [g , g (j−1) ].
If g (s+1) = {0} and g (s) = {0}, then g is said to be nilpotent of step s. A Lie group G is nilpotent of step s whenever its Lie algebra is nilpotent of step s. Let exp : g → G be a exponential map, and G be a connected and simplyconnected nilpotent Lie group with Lie algebra g. Then, exponential map exp is a diffeomorphism from g to G. Let A be a diagonalisable linear operator on g with positive eigenvalues, and µ > 0. Define the mappings are of the form Then, let us give a family of dilations of a Lie algebra g as follow which satisfies: (i) δ µ is a morphism of the Lie algebra g, that is, a linear mapping from g to itself which respects to the Lie bracket: Remark 2.1. (i) If a Lie algebra g admits a family of dilations, then it is nilpotent, but not all nilpotent Lie algebras admit a dilation structure.
(ii) Since the exponential mapping exp is a global diffeomorphism from g to G, it induces the corresponding family on G which we may still call the dilations on G and denote by δ µ . Thus, for x ∈ G we will write δ µ (x) or abbreviate it writing simply µx, and the origin of G will be usually denoted by 0.
Definition 2.1. Let δ µ be a dilations on G. We say that a Lie group G is a homogeneous Lie group if: (a) It is a connected and simply-connected nilpotent Lie group G whose Lie algebra g is endowed with a family of dilations {δ µ }.

Now, we give some two examples of homogeneous groups.
Example 2.1. The Euclidean space R N is a homogeneous group with dilation given by the scalar multiplication.
Example 2.2. If N is a positive integer, the Heisenberg group H N is the group whose underlying manifold is C N × R and whose multiplication is given by The mappings {δ µ } give the dilation structure to an N -dimensional homogeneous Lie group G with where (x 1 , · · · , x N ) are the exponential coordinates of x ∈ G, d j ∈ N for every j = 1, 2, · · · , N and 1 = d 1 = · · · = d m < d m+1 ≤ · · · ≤ d N for m := dim(V 1 ). Here the group G and the algebra g are identified through the exponential mapping. It is customary to denote with Q := k i=1 i · dim(V i ) the homogeneous dimension of G which corresponds to the Hausorff dimension of G. From now on Q will always denote the homogneous dimension of G. For example, the homogeneous dimension of Heisenberg group H N is Q := 2N + 2. Now, we define a homogeneous quasi-norm on a homogeneous Lie group G to be a continuous function g : G → R + with the following properties: be the quasi-ball of radius r > 0 about x with respect to the homogeneous quasinorm g. Then, we have that It can be noticed that B g (x, r) is the left translate by x of B g (0, r), which in turn is the image under δ r of B g (0, 1). Moreover, let be the unit sphere with respect to the homogeneous quasi-norm g. Then there is a unique positive Radon measure σ on S g (0) such that for all f ∈ L 1 (G), we have Let G be a homogeneous Lie group, with its basis X 1 , . . . , X N , generating its Lie algebra g through their commutators. Then, the sub-Laplacian operator is defined as L := X 2 1 + · · · + X 2 N . In the sequel, we use the following notations for the horizontal gradient and for the horizontal divergence Using the Green's first and second formulae, we can define the p-sub-Laplacian on homogeneous groups G as Recently, a great deal of attention has been focused on studying of equations or systems involving fractional Laplacian and corresponding nonlocal problems, both for their interesting theoretical structure and for their concrete applications, see [6,17,19,20] and references therein. The fractional p-Laplacian operator (−∆ p ) s , s ∈ (0, 1), is defined as This type of operator arises in a quite natural way in many different contexts, such as, the thin obstacle problem, finance, phase transitions, anomalous diffusion, flame propagation and many others. Let G be a homogeneous Lie group with homogeneous dimension Q, p > 1 and s ∈ (0, 1). Compared to the fractional p-Laplacian problem, the fractional p-sub-Laplacian (−∆ p,g ) s on G can be defined as where g is a quasi-norm on G and B g (x, ε) is a quasi-ball with respect to g, with radius ε centered at x ∈ G . Now we recall the definitions of the fractional Sobolev spaces on homogeneous Lie groups G. For a measurable function u : G → R we define the Gagliardo quasi-seminorm by For p > 1 and s ∈ (0, 1), we introduce the the functional Sobolev space on homogeneous Lie groups G by W s,p g (G) = {u ∈ L p (G) | u is measurable and [u] s,p,g < +∞}, and endowed with the norm Similarly, if Ω ⊂ G is a Haar measurable set, we define the Sobolev space u is measurable and In [16, Theorem 2], Kassymov and Suragan given the following analogue of the fractional Sobolev inequality on homogeneous groups G.
Theorem 2.1. Let G be a homogeneous group with homogeneous dimension Q. Assume that p > 1, s ∈ (0, 1), Q > sp and g denotes a quasi-norm on G. Then, for any measurable and compactly supported function u : G → R, there exists a positive constant C = C(Q, p, s) > 0 such that where p * s := p * (Q, s, p) = Qp Q−sp is the fractional critical Sobolev exponents on homogeneous group.

2.2)
Indeed, if u Eg = 0, we get that u L p (Ω) = 0 and Q |u(x)−u(y)| p g(y −1 •x) Q+sp dxdy = 0. Then, the above equalities imply that u = 0 a.e. in Ω and u(x) = u(y) = const. a.e. in Q. So, we can get that u = 0 a.e. in G and · Eg is a norm on E g .
Let E 0 g = {u ∈ E g : u = 0 in G\Ω} be a subspace of E g . Then, for any p > 1, E 0 g is a Banach space and have the following properties. Lemma 2.1. The following hold.
Thus u W s,p g (Ω) ≤ u Eg and deduced the result (i). (ii) For each u ∈ E 0 g , we get u = 0 on G \ Ω. Hence, u L 2 (G) = u L 2 (Ω) and Assume that p > 1, s ∈ (0, 1), Q > sp and g be a quasi-norm on G. Then for every u ∈ E 0 g there exists a positive constant c = c(Q, s) > 0 depending on Q and s such that Proof. For any u ∈ E 0 g , by Lemma 2.1 (ii) and Theorem 2.1, we know that u ∈ W s,p g (G) and W s,p g (G) → L p * s (G). Then, we have and completes the proof of Theorem 2.2.
g be a Cauchy sequence. By Lemma 2.1 and Theorem 2.2, {u k } k is Cauchy sequence in L p (Ω) and so {u k } k has a convergent subsequence. We assume u k → u in L p (Ω). Since u k = 0 in G \ Ω, we define u = 0 in G \ Ω and then u k → u strongly in L p (G) as k → ∞. So, there exists a subsequence of {u k } k , still denoted by {u k } k , such that u k → u a.e. in G. By Fatou's Lemma and using the fact that {u k } is a Cauchy sequence, we get that u ∈ E 0 g and u k − u Eg → 0 as k → +∞. Hence E 0 g is a Banach space. Reflexivity of E 0 g follows from the fact that E 0 g is a closed subspace of reflexive Banach space W s,p g (G).
From above results, we can defined the following scalar product and norm for the reflexive Banach space E 0 g . Since u = 0 a.e. in G\Ω, we note that the (2.5) and (2.6) can be extended to all G. Moreover, for any u ∈ E g , by Theorem 2.2 and the embedding L p * s (Ω) → L p (Ω), there exist C 1 and C 2 > 0 such that This imply that the norm · E 0 g on E 0 g is equivalent to the norm · Eg on E g , and the norm · E 0 g involves the interaction between Ω and G \ Ω. But the norms in (2.1) and (2.2) are not same because Ω × Ω is strictly contained in Q.
g , by Lemmas 2.1 and (2.7), {u k } k is bounded in W s,p g (Ω) and also in L p (Ω). Then by assumption on Ω and [4, Corollary 7.2], there exists u ∈ L q (Ω) such that up to a subsequence u k → u in L q (Ω) as k → ∞ for any q ∈ [1, p * s ). Since u k = 0 on G \ Ω, we can define u := 0 in G \ Ω and we get u k → u in L q (G). From Theorem 2.2, Lemma 2.1 and Lemma 2.3, we have that the embedding E 0 g → L q (Ω) is continuous for any q ∈ [1, p * s ] and compact whenever q ∈ [1, p * s ). Let S p * s be the best constant for the embedding of E 0 g → L p * s (Ω) defined by 3. The fibering properties. Since the energy functional I λ is not bounded below on the space E 0 g , we consider the Nehari minimization problem: for λ > 0, where N λ := u ∈ E 0 g \{0} : I λ (u), u = 0 . For any u ∈ N λ , the following equality hold which implies that N λ contains all nonzero solutions of equation (1.1). Moreover, we have the following result.
Lemma 3.1. I λ is coercive and bounded below on N λ for all λ > 0.
Lemma 3.2. For each λ > 0, let u 0 be a local minimizer for I λ on N λ \N 0 λ , then u 0 is a critical point of I λ .
Proof. Since u 0 is a local minimizer for I λ on N λ , that is, u 0 is a solution of the optimization problem min{I λ (u) : u ∈ N λ } = min{I λ (u) : Φ λ (u) = 0}.
Then, by the theory of Lagrange multipliers, there exists a constant θ ∈ R such that , u 0 = 0, thus θ = 0, this completes the proof. Remark 3.1. Lemmas 3.1 and 3.2 imply that the functional I λ is bounded below on an appropriate subset of E 0 g and the minimizers of functional I λ on subsets N + λ , N − λ giving raise to solutions of (1.1). Now, for t > 0, define the fibering maps φ u : t → I λ (tu) as We note that This gives that tu ∈ N λ if and only if φ u (t) = 0 and in particular, u ∈ N λ if and only if φ u (1) = 0. Moreover, for each u ∈ N λ , from (3.1), (3.3) and (3.5) we get that which and (3.4) yield that N + λ , N − λ and N 0 λ are corresponding to local minima, local maxima and points of inflection of φ u (t), namely . In order to understand the Nehari manifold and the fibering maps, we consider the function m u : R + → R defined by Clearly, for any t > 0, (3.6) Thus, we obtain that From the expression (3.5) of φ u , we see that the behavior of the fibering maps φ u according to the sign of Ω f (x)|u| α dx and Ω h(x)|u| β dx, then we will study the following four cases.