Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schr\"{o}dinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}u+V(x)m(u)=f(x,u),\ x\in\mathbb{R}^{d},$$ where $0<s<1$, $d\geq2$ and $(-\triangle)^{s}_{m}$ is the fractional $M$-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.


Introduction and main result
In this paper, we are concerned with the study of the fractional M -Laplacian equation: where (−△) s m is the fractional M -Laplace operator, 0 < s < 1, d ≥ 2, m : R → R is an increasing homeomorphism, V : R d → R and f : R d × R → R are given functions.
In the last years, problem (1.1) has received a special attention for the case s = 1 and m(t) = t, that is, when it is of the form We do not intend to review the huge bibliography related to the equations like (1.2), we just emphasize that the potential V : R d → R has a crucial role concerning the existence and behaviour of solutions. For example, when V is radially symmetric, it is natural to look for radially symmetric solutions, see [36,42]. On the other hand, after the paper of Rabinowitz [33] where the potential V is assumed to be coercive, several different assumptions are adopted in order to obtain existence and multiplicity results (see [6,9,22,39,40]). For the case s = 1, problem (1.1) becomes where the operator △ m u = div(m(|∇u|)|∇u|) named M -Laplacian. This class of problems arises in a lot of applications, such as, Nonlinear Elasticity, Plasticity, Generalized Newtonian Fluid, Non-Newtonian Fluid, Plasma Physics. The reader can find more details involving this subject in [2,11,27,28] and the references therein.
Notice that when 0 < s < 1 and m(t) = |t| p−2 t, p ≥ 2, problem (1.1) gives back the fractional Schrödinger equation where (−△) s p is the non-local fractional p-Laplacian operator. The literature on non-local operators and on their applications is quite large. We can quote [7,17,18,34,35] and the references therein. We also refer to the recent monographs [17,30] for a thorough variational approach of non-local problems. In the last decade, many several existence and multiplicity results have been obtained concerning the equation (1.3), (see [5,19,38]). In [10], the authors studied the existence of multiple solutions where the nonlinear term f is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. In [4], Vincenzo studied the existence of infinitely many solutions for the problem (1.3), when f is superlinear and V can change sign.
Contrary to the classical fractional Laplacian Schrödinger equation that is widely investigated, the situation seems to be in a developing state when the new fractional M -Laplacian is present. In this context, the natural setting for studying problem (1.1) are fractional Orlicz-Sobolev spaces. Currently, as far as we know, the only results for fractional Orlicz-Sobolev spaces and fractional M -Laplacian operator are obtained in [3,8,12,13,14,31,37]. In particular, in [12], Bonder and Salort define the fractional Orlicz-Sobolev space associated to an N -function M and a fractional parameter 0 < s < 1 as  Motivated by these above results, our first aim is to prove the compact embedding W s,M (Ω) ֒→ L M * (Ω) where M * is the Sobolev conjugate of M and Ω is bounded. Furthermore, we state the continuous embedding W s,M (R d ) ֒→ L M * (R d ). Hence the compact embedding W s,M (Ω) ֒→ L Φ (Ω) and the continuous embedding W s,M (R d ) ֒→ L Φ (R d ) remain true for any N -function Φ such that M * is essentially stronger than Φ. (see Definition 2.1).
Our next aim is to study the existence and the multiplicity of nontrivial weak solutions of problem (1.1), where the new fractional M -Laplacian is present. Under suitable conditions on the potentials V and f (will be fixed bellow), we deal with a new compact embedding theorem on the whole space R d . Also we establish some useful inequalities which yields to apply a variant of Fountain theorem due to Zou [41]. As far as we know, all these results are new.
Related to functions m, M, V and f , our hypotheses are the following: Conditions on m and M : Conditions on V : Conditions on f : 1. m(t) = q|t| q−2 t, for all t ∈ R, with 2 < q < d (also satisfies condition (M 3 )).
Under the above hypotheses, we state our main results. is compact.
The boundedness of Ω in Theorem 1.2 is a natural requirement for the compactness theorem, but, as we shall show in the next theorem, not necessary for the continuous embedding.
is continuous.
2. Moreover, for any N -function B such that B ≺≺ M * , the embedding is continuous.
In studying the existence of solution of problem (1.1), it is common to relax the notion of solution by considering weak solutions. By these we understand functions in W s,M (Ω) that satisfy (1.1) in sense of distribution.
This paper is organized as follows. In Section 2, we give some definitions and fundamental properties of the spaces L M (Ω) and W s,M (Ω). In Section 3, we prove Theorems 1.2 and 1.3. In Section 4, we introduce our abstract framework related to problem (1.1). Finally, in Section 5, using a variant Fountain theorem [41], we prove Theorem 1.4.

Orlicz spaces
where m : R → R is non-decreasing, right continuous, with m(0) = 0, m(t) > 0 for all t > 0 and m(t) → ∞ as t → ∞ (see [24], page 9). We call the conjugate function of M , the function denoted M and defined by where m : R → R, m(t) = sup{s : m(s) ≤ t}. We observe that M is also an N -function and the following Young's inequality holds true st ≤ M (s) + M (t) for all s, t ≥ 0, (see [1], page 229).  In what follows, we say that an N -function M satisfies the △ 2 -condition, if for some constant K > 0. This condition can be rewritten in the following way For each s > 0, there exists K s > 0 such that , for all t ≥ 0, (see [24], page 23). (2.10) Definition 2.1. Let A and B be two N -functions, we say that A is essentially stronger than B, B ≺≺ A in symbols, if for each a > 0 there exists x a ≥ 0 such that The previous definition 2.1 is equivalent to, lim t→+∞ B(kt) A(t) = 0, for all positive constant k (see [32], Theorem 2).
The Orlicz spaces Hölder's inequality reads as follows: (see [25], Theorem 4.7.5) In the following, we recall a few results which will be useful in the sequel.

Fractional Orlicz-Sobolev spaces
In this subsection we give a brief overview on the fractional Orlicz-Sobolev spaces studied in [12], and the associated fractional M -laplacian operator.
This space is equipped with the norm, A variant of the well-known Frèchet-Kolmogorov compactness theorem gives the compactness of the embedding of W s,M into L M . is compact.
We recall that the fractional M -Laplacian operator is defined as where P.V is the principal value. This operator is well defined between W s,M (R d ) and its dual space W −s,M (R d ). In fact, in [[12], Theorem 6.12] the following representation formula is provided

Embedding Theorems
After the above brief review, we are able to prove our main results involving the fractional Orlicz-Sobolev spaces.

Proof of Theorem 1.2
The proof will be carried out in several lemmas. we start by establishing an estimate for the Sobolev conjugate N -function M * defined by (2.8).
2. For every ǫ > 0, there exists a constant K ǫ such that for every t, Proof. Let u ∈ W s,M (Ω), then there exists θ > 0 such that . This ends the proof.

Lemma 3.3.
Let Ω be a bounded subset of R d with C 0,1 -regularity and bounded boundary. Let M be an N -function satisfying condition (m 1 ). Then, given 0 < s ′ < s < 1, it holds that the embedding is continuous.
Proof. We closely follow the method employed in [ [13], Proposition 2.9]. The normalization condition M (1) = 1 is by no means restrictive. From Lemma 2.4 it is inferred that, where δ is the diameter of Ω. To estimate the second term, we invoke (3.20) and we obtain By homogeneity of the seminorm [.] s ′ ,1 , we obtain On the other hand, since Ω is bounded, there exist C > 0 such that Proof of theorem 1.2. Let u ∈ W s,M (Ω) \ {0} and suppose for the moment that u is bounded on Ω. Then λ → Ω M * (|u(x)|/λ)dx decreases continuously from infinity to zero as λ increases from zero to infinity. So that By the definition of the norm (2.11), we see that k = u (M * ) . Let So there is a constant C 1 > 0 such that On one hand, by (3.18) and the Hölder inequality, for ǫ = 1 2C 1 , we have where On the other hand, since ω is Lipschitz continuous, there exists K > 0 such that, where C 4 = KC 1 C 3 . Combining (3.26) and (3.27), we obtain from which it follows that where To extend (3.28) to arbitrary u ∈ W s,M (Ω), let Clearly f n is 1-Lipschitz continuous function. By Lemma 3.2, (u n ) belongs to W s,M (Ω). So in view of (3.28) On the other hand, we have Let k n = u n (M * ) , the sequence (k n ) is non-decreasing and converges in view of (3.31). Put k ′ = lim n→+∞ k n , by Fatou's Lemma we get Thus the first assertion of the theorem is proved. Now, let's turn to the compactness embedding.
In what follows we show that Let i ∈ N, we distingue two cases:

Combining (3.34) and (3.35), we obtain
. Then, by using Proposition 2.3, we infer and for all N -function B ≺≺ M * , we have The proof of Theorem 1.3 is completed.

Variational setting of problem (1.1) and some useful tools
In this section, we will first introduce the variational setting for problem (1.1). In view of the presence of potential V , our working space is where We define the functional G : E → R by (4.37) We consider the following family of functionals on E Lemma 4.1. The functional I λ is well defined on E, moreover I λ ∈ C 1 (E, R) and for all v ∈ E, Definition 4.2. We say that u ∈ E is a weak solution to (1.1) if u is critical point of I 1 , which means that u satisfies I Lemma 4.3. Assume that (m 1 ) and (V 1 ) are satisfied. Then, the following properties hold true: Proof. The proof of the first assertion is given by [ , , for x ∈ R d .
From the definition of the norm (2.11), we deduce that M) ).
On the other hand, let ǫ > 0, β = u (V,M) − ǫ in Lemma 2.4, we get Letting ǫ → 0 in the above inequality, we obtain Thus the assertion (ii) and the proof of Lemma 4.3 is complete.
Proof. Using Lemma 2.4, we have, for all λ > 0 and t > 1, Again with Lemma 2.4, we get Thus Ψ • Φ ≺≺ M and the proof is completed. Now we state our embedding compactness result. Under the assumption (m 1 ), (V 1 ) and (V 2 ), the embedding from E into L Φ (R d ) is compact.
Proof. Let Φ be an N -function satisfying (4.39) such that Φ ≺≺ M and (v n ) be a bounded sequence in E, since E is reflexive, up to subsequence, v n ⇀ v in E. Let u n = v n − v, u n ⇀ 0 in E. We have to show that u n → 0 in L Φ (R d ), by Proposition 2.2 this means that R d Φ(u n )dx → 0, as n → +∞. (4.42) According to Vitali's theorem it suffices to show that, the sequence (Φ(u n )) is equi-integrable, which means: We do the proof in two steps. We start by checking (a). Let L > 0 and
On the other hand, the following limit holds then, for all ǫ > 0, there exists δ ǫ > 0 such that if |t| ≤ δ ǫ , we have By [ [25], Proposition 4.6.9], we know that . Therefore, for all measurable subset of R d such that meas(B) ≤ δ ǫ , We conclude that (Φ(u n )) is uniformly integrable and tight over R d . Thus the Lemma 4.5 is proved.
Corollary 4.6. Under (m 1 ) and (M 1 ), the embedding from E into L µ (R d ) is compact.
Lemma 4.7. Assume that (m 1 ) and (M 1 ) are satisfied. Then the functional A is weakly lower semicontinuous on E.
Proof. By [ [8], Lemma 3.3], G is weakly lower semi-continuous, so it is enough to show that Ψ is too. Let (u n ) ⊂ E be a sequence which converges weakly to u in E. Since E is compactly embedded in L µ (R d ), it follows that (u n ) converges strongly to u in L µ (R d ). Up to a subsequence, Using Fatou's lemma, we get Therefore, A is weakly lower semi-continuous. Thus the proof. Proof. Let u, v ∈ E. Using Hölder and Young inequalities, we compute then according to Lemma 2.4, we obtain combining [Lemma 2.9, [12]], Lemma 4.3 and Corollary 4.6, we get From the last inequality, we conclude that I ′ λ maps bounded sets to bounded sets for λ ∈ [1, 2].
Lemma 4.9. If u n ⇀ u in E and A ′ (u n ), u n − u → 0, as n → +∞, (4.50) Proof. Since (u n ) converges weakly to u in E, then ([u n ] (s,M) ) and ( u n (V,M) ) are a bounded sequences of real numbers. That fact and relations (i) and (ii) from lemma 4.3, imply that the sequences (G(u n )) and (Ψ(u n )) are bounded. This means that the sequence (A(u n )) is bounded. Then, up to a subsequence, A(u n ) → c. Furthermore, Lemma 4.7 implies Therefore, combining (4.50), (4.51) and (4.52), we conclude that A(u) = c.
Taking into account that u n + u 2 converges weakly to u in E and using again the weak lower semi-continuity of A, we find We argue by contradiction, and suppose that (u n ) does not converge to u in E. Then, there exists β > 0 and a subsequence (u nm ) of (u n ) such that

by (i) and (ii) in lemma 4.3, we infer that
On the other hand, the △ 2 −condition and relation (M 2 ) enable us to apply [ [26], Theorem 2.1], in order to obtain Letting m → ∞ in the above inequality, we get That is a contradiction. It follows that (u n ) converges strongly to u in E. Thus lemma 4.9 is proved.

Proof of Theorem 1.4
Let (E, . ) be a Banach space and E = j∈N X j with dim X j < ∞ for any j ∈ N.
Consider a C 1 -functional I λ : E → R defined as Let, for k ≥ 2, In order to prove Theorem 1.4, we apply the following variant of fountain Theorem due to Zou [41].
Theorem 5.1. Assume that I λ satisfies the following assumptions: (i) I λ maps bounded sets to bounded sets for λ ∈ [1,2] and Then there exist λ n → 1, u λn ∈ Y n such that Particularly, if (u λn ) has a convergent subsequence for every k, then I 1 has infinitely many nontrivial critical points {u k } ∈ E\{0} satisfying I 1 (u k ) → 0 − as k → ∞.
Since E is reflexive and separable, we choose a basis {e j : j ∈ N} of E and {e * j : j ∈ N} of E * such that e * i , e j = δ i,j , ∀i, j ∈ N. Let X j = e j for all j ∈ N and In order to apply Theorem 5.1, we need the following lemmas. Proof. Using Lemma 4.8, we observe that (I ′ λn (u λn )) n∈N is bounded in E * . As E = ∪ n Y n , we can choose w n ∈ Y n such that w n → u 0 as n → +∞. Proof. Evidently B(u) ≥ 0 for all u ∈ E follows by (f 1 ). We claim that for any finite dimensional subspace H ⊂ E, there exists a constant c H > 0 such that We argue by contradiction, suppose that for any n ∈ N there exists u n ∈ H \ {0} such that For each n ∈ N, let v n = u n u n ∈ H, v n = 1, then Up to a subsequence, we may assume that v n → v for some v ∈ H and v = 1 Furthermore, there exists a constant δ 0 > 0 such that In fact, if not, Let m > n, then A n ⊂ A m and meas(A n ) ≤ meas(A m ) < 1 m → 0 as m → +∞, it yields This together with (f 1 ) yields v = 0, a.e. which is in contradiction to v = 1. Thus (5.59) is proved.
By Hölder inequality and Corollary 4.6, it holds that (5.60) and for all n ∈ N, Taking into account (5.58) and (5.59), for n large enough, we get Therefore, for n large enough, we obtain which contradicts (5.60). Thus the claim.
By (5.57), we have This implies that B(u) → ∞ as u → ∞ in H. The proof of Lemma 5.3 is complete.
Lemma 5.4. Let (l k ) k∈N the sequence defined by Proof. It is clear that (l k ) is non-increasing positive sequence. So there exists z ≥ 0 such that l k → z as k → +∞. For any k ∈ N, there exists u k ∈ Z k such that u k = 1 and u k L µ (R d ) ≥ l k 2 . We observe that u k ⇀ u in E and e * n , u k = 0 for k > n. So e * n , u = lim k→+∞ e * n , u k = 0, for all n ∈ N, which gives u = 0. Corollary 4.6 implies that u k → 0 in L µ (R d ). Thus z = 0. Proof. Using Lemma 4.3 and Hölder inequality, for any u ∈ Z k and λ ∈ [1, 2], Combining (5.61) and (5.63), we get Choose θ > 0 (θ will be fixed later) and Besides, by (5.64), for each k ∈ N, we have The proof of Lemma 5.5 is complete.
Thus the proof.
Proof of Theorem 1.4: Since I λ (u) ≤ I 1 (u) for all u ∈ E and I 1 maps bounded sets to bounded sets, we see that I λ maps bounded sets to bounded sets uniformly for λ ∈ [1,2]. Moreover, I λ is even. Then the condition (i) in Theorem 5.1 is satisfied. Besides, Lemma 5.3 shows that the condition (ii) in Theorem 5.1 holds. While Lemma 5.5 together with Lemma 5.6 implies that the condition (iii) holds.
Claim 3: We claim that the sequence (u n ) n∈N is bounded in E.
In fact, if u n ≤ 1, for all n ∈ N, nothing to prove. If not, we define the following sets:  M) ).
for some constants C 1 , C 2 > 0. Since p < m 0 , there exists D 1 > 0 such that [u n ] (s,M) ≤ D 1 for all n ∈ N 2 . It follows that u n ≤ 1 + D 1 for all n ∈ N 2 . (5.73) By the same argument as above, we can see that for some D 2 > 0, u n (V,M) ≤ D 2 for all n ∈ N 3 . Then u n ≤ 1 + D 2 for all n ∈ N 3 . By accumulating all the preceding cases (5.72), (5.73), (5.74) and (5.75), we deduce that the sequence (u n ) n∈N is bounded in E.

Claim 4:
The sequence (u n ) admits a strongly convergent subsequence in E.
In fact, in view of Claim 3 and up to subsequence, u n ⇀ u 0 as n → +∞, for some u 0 ∈ E. On one hand, according to Lemma 5.  On the other hand, by Hölder inequality and Lemma 4.5, we get [m(u n ) − m(u 0 )](u n − u 0 )dx → 0, as n → ∞.
According to Lemma 4.9, (u n ) converges strongly to u 0 in E. Thus the claim. Now by the last assertion of Theorem 5.1, we conclude that I 1 has infinitely many nontrivial critical points. Therefore, (1.1) possesses infinitely many nontrivial solutions. The proof of Theorem 1.4 is complete.