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Least-energy nodal solutions of nonlinear equations with fractional Orlicz-Sobolev spaces

In our work, we prove the existence of least-energy nodal solutions for nonlinear equations in which the new fractional Orlicz Laplacian is present. Precisely, we prove a compact embeddings result for weighted fractional Orlicz-Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show our desired result.


Introduction
In this paper, we are concerned with the existence of leat-energy nodal (sign-changing) weak solutions for the following problem where (−△ g ) α stands the fractional g−Laplace operator, N ≥ 3, α ∈ (0, 1) and g : R → R will be specified later, while f and K are generally two continuous functions whose assumptions will be introduced in what follows.
The novelty of our work (see Theorem 2.9) is the fact that we combine several different phenomena in one problem. The features of this paper are the following: (1) the existence of nodal solutions.
(2) the presence of the new fractional g−Laplacian.
(3) the lack of compactness due to the free action of the translation group on R N .
To the best of our knowledge, this is the first paper proving the existence of sign-changing solutions with the combined effects generated by the above features.
Recently, non-local problems and operators have been widely studied in the literature and have attracted the attention of lot of mathematicians coming from different research areas. This type of non-local operators arises in the description of various phenomena in the applied sciences, such as optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, ultra-relativistic limits of quantum mechanics... (see, [24]) and the references therein. More recently, the papers of Caffarelli et al. [19,20,21], led to a series of works involving the fractional diffusion, (−△) α , 0 < α < 1 (see, for instance, [5,15,16,18,24,25,38,39]). In the last references, the authors tried to see which results survive when the Laplacian is replaced by the fractional Laplacian.
A natural question is to see what results can be recovered when the standard p(x)−Laplace operator is replaced by the fractional p(x)−Laplacian. To our best knowledge, Kaufmann et al. [33] firstly introduced some results on fractional Sobolev spaces with variable exponent W α,q(.),p(.,.) (Ω) and the fractional p(x)−Laplacian. There, the authors established compact embedding theorems of these spaces into variable exponent Lebesgue spaces. In [10], Bahrouni and Rȃdulescu obtained some further qualitative properties of the fractional Sobolev space W α,q(.),p(.,.) (Ω) and the fractional p(x)−Laplacian. After that, some studies on this kind of problems have been performed by using different approaches, see [4,7,31,41].
The study of nonlinear elliptic equations involving quasilinear homogeneous type operators is based on the theory of Sobolev spaces in order to find weak solutions. In the case of nonhomogeneous differential operators, the natural setting for this approach is the use of Orlicz-Sobolev spaces. For more details on the theory of Orlicz and Orlicz-Sobolev spaces, we can cite [28,34,36] and the references therein. It is, therefore, a natural question to see what results can be recovered when the g-Laplace operator is replaced by the fractional g-Laplacian.
It is worth mentioning that there are some papers concerning related equations involving the fractional g−Laplace operator and fractional Orlicz-Sobolev spaces. In fact, results on this subject are few. In 2019, J. Fernández. Bonder firstly introduced some results on this context (see [17]). In particular, the authors generalize the g−Laplace operator to the fractional case. They also introduce a suitable functional space to study an equation in which a fractional g−Laplace operator is present. After that in 2020 some authors, like S. Bahrouni and A. M. Salort have continued these studies see [6,8,9,37]. More precisely, they proved some basic results like embedding problems and some other fundamental properties. As we mentioned above, the literature survey on problems involving the fractional g−Laplacian is almost meagre since it is still a work in progress. To the best of our knowledge, there is no paper devoted to the studies of the existence of sign-changing solution for the equations contains the fractional g−Laplacian.
When α = 1 and g(t) = |t| p−2 t, Eq. (P) gives back the classical quasilinear equation We do not intend to review the huge bibliography of nodal solutions for the above equations, we just emphasize that the nodal Nehari manifold approach is a very efficient method to prove the existence of sign-changing solutions, see [11,12,13,23,26,30,40]. In [12], S. Barile and G. M. Figueredo have studied the following equation: where N ≥ 3, 2 ≤ p < N, a, b, f are C 1 real functions and V, K are continuous positives functions. Under some further assumptions on f, a, b, K and V , and using the nodal Nehari manifold method, they proved the existence of a sign-changing solution with two nodal domains. In [26], G. M. Figueredo considered the following equation: where Ω is a bounded domain in R N , M is C 1 function, f is a superlinear C 1 class function with subcritical growth and ∆ g is the Orlicz g-Laplace operator. By using the nodal Nehari manifold method, he proved the existence of a nodal solution.
To the best of our knowledge, there are no results concerning the existence of sign-changing solutions for the equations in which the fractional g−Laplace operator is present. Hence, a natural question is whether or not there exist nodal solutions of problem (P). The goal of the present paper is to develop a thorough qualitative analysis in this direction. Our method is inspired by the work of S. Barile and G. M. Figueredo [12].
The paper is organized as follows. In Sect. 2, we give some definitions and fundamental properties of Orlicz-Sobolev spaces and fractional Orlicz-Sobolev spaces. In Sect. 3 we show a compact embedding type result. The last section is divided into two parts: In the first subsection, we prove some technical lemmas which will be useful in the sequel. In the second part, we conclude the proof of our main result (see Theorem 2.9).

Mathematical background and hypotheses
In this section, we recall some necessary properties about Orlicz-Sobolev and fractional Orlicz-Sobolev spaces. For more details we refer the readers to [34,36,37]. We start by recalling some definitions about the well-known N -functions. Let g be a real-valued function defined on R and having the following properties: (g 1 ) (a) g(0) = 0, g(t) > 0 if t > 0 and lim t→+∞ g(t) = +∞.
(b) g is nondecreasing and odd function.
(c) g is right continuous.
The real-valued function G defined on R by is called an N -function. It is easy to see that G is even, positive, continuous and convex function. Moreover, one has G(0) = 0, t → 0 as t → 0 and G(t) t → +∞ as t → +∞. The conjugate N -function of G, is defined bỹ In what follows, we assume that the N -functionsG satisfies the △ 2 -condition and we suppose that G and g are of class C 1 and satisfy the two following conditions (g 2 ) There exist L and m such that L ∈ ( N 2 , N ), and L < m < min{ L The assumption (g 2 ) implies that G satisfies the △ 2 -condition.
The Orlicz space L G (R N ) is the vectorial space of measurable functions u : R N → R such that [1,29]).
Next, we introduce the Orlicz-Sobolev spaces and the fractional Orlicz-Sobolev spaces. We denote by W 1,G (R N ) the Orlicz-Sobolev space defined by is a Banach space with respect to the norm We denote by W α,G (R N ) the fractional Orlicz-Sobolev space defined by where 0 < α < 1 and ρ(u) = Since G andG satisfy the △ 2 -condition, then the fractional Orlicz-Sobolev spaces is a reflexive separable Banach space. We define another norm on W α,G (R N ) whereρ(u) = ρ(u) + ρ(u). The two norms u and u α,G are equivalent. More precisely, we have (see [8] Lemma 3.5 page 11).
Another N -function related to function G, is the Sobolev conjugate function G * defined by is compact.
2. If B is an N -function satisfies the △ 2 -condition such that B ≺≺ G * (G * is given in (2.8)), then the embedding is continuous. In particular, we have the continuous embeddings and The fractional Orlicz g-Laplacian operator is defined as where P.V. is the principal value. The operator (−△) α g is well defined between W α,G (R N ) and its dual space W −α,G (R N ). In fact, we have that Under the assumptions (g 1 ) − (g 3 ), some elementary inequalities and properties listed in the following lemmas are valid.

From
We make the following assumptions on functions f and K.
• The function f : R −→ R is of class C 1 and satisfy the following growth conditions at the origin and infinity • The function K : R N −→ R is continuous on R N and satisfy the following assumptions: If (A n ) n ⊂ R N is a sequence of Borel sets such that the Lebesgue measure mes(A n ) ≤ R, for all n ∈ N and some R > 0, then lim r→+∞ An∩B c r (0) K(x) dx = 0, uniformly in n ∈ N.
As a consequence of the above assumptions, we can deduce the following lemma which will be useful in the sequel.
t is even and increasing, for all t > 0.
Our main result asserts that problem (P) has at least one sign-changing solution. More precisely: are fulfilled, then, problem (P) admits at least one nodal nontrivial weak solution.

Compactly embedding
In this section, we are going to prove a compactness embedding result. Let M be an N -function, we define the weighted Orlicz space and then, for fixing ε > 0, there exist 0 < t λ 0 (ε) < t λ 1 (ε) and C(ε, λ) > 0, such that Using (3.18), assumption (K 1 ) and integrating over B c r (0) where r > 0, we have Evidently, the sequence (v n ) n is bounded in W α,G (R N ). Then, applying Theorem 2.1 and Lemma 2.6 , there is C > 0 such that We observe that mes(A n ) < +∞ for all n ∈ N, indeed: By assumption (K 2 ), for all ε (3.20) Using (3.19), and choosing ε in (3.20), we can see that In the other side, from Theorem 2.1 and the fact that K is a continuous function, we get lim n→+∞ Br(0) Putting together (3.21) and (3.22), we find that Now, we prove a compactness embedding results related to the non-linear term.
(3) Since |t ± − s ± | ≤ |t − s| for all t, s ∈ R, the proof is similar to the first case.

Technical lemmas and proof of main result
We divide this section into two parts. In the first, we give some technical lemmas. In the second part, using the results obtained in the first subsection, we prove Theorem 2.9.
Again, we recall that we look for sign-changing weak solutions of problem (P ), that is a function u ∈ W α,G (R N ) such that u + := max{u, 0} = 0, u − := min{u, 0} = 0 in R N and In view of assumptions on K and f , we see that J is Frèchet differentiable and Note that M = ∅ (will be treated later in Lemma 4.7 ).
We look for a least energy sign-changing weak solution of problem (P), it means to look for function w ∈ M such that (1) Using Lemma 2.2 and taking into account the fact that G(t) is an even and increasing function, we see that Q t,s is positive, for all (t, s) ∈ [0, +∞) × [0, +∞).
(2) By using the fact that g(t) is an odd and nondecreasing function, we prove that E t,s and H t,s are positives, for all (t, s) ∈ [0, +∞) × [0, +∞).
In the following, we give some technical Lemmas which will be useful later.

Technical Lemmas
In this subsection we give some technical lemmas which will be useful in the proof of our main result.
On the other hand Combining the above pieces of informations, we conclude that Thus the proof of (i). The proofs of (ii) and (iii) are similar to that in (i).    In the other hand, since w ∈ N , one has Combining (4.29) and (4.30), we obtain where By Lemma 2.8, it follows that I 1 , I 2 , I 3 , I 4 , I 5 > 0, this gives the desired result.
In the following lemma, we prove that J is coercive on N and in particular on M. (ii) There exists ̺ > 0 such that u ≥ ̺ for all u ∈ N and w ± ≥ ̺ for all w ∈ M.
Lemma 4.6. Let {w n } n ⊂ M such that w n ⇀ w in W α,G (R N ), then w ± = 0.
Proof. By Lemma 4.5, there exists ̺ > 0 such that w ± n ≥ ̺, for all n ∈ N. According to Lemma 4.3 and have in mind that w n ∈ M, we obtain J ′ (w ± n ), w ± n ≤ J ′ (w n ), w ± n = 0. Applying Lemma 2.7, we infer that Putting together (4.34) and (4.35), we deduce that On the other hand, according to Lemma 3.2-(3), we have that Combining (4.36) with (4.37), we get thus, we conclude that w ± = 0. Thus we prove the desired result.
In the following, we are able to prove that every nontrivial sign-changing function in W α,G (R N ) corresponds to a suitable function in M.
Invoking Lemma 2.7, (4.39) and (K 1 ), we deduce, for t and s large enough, that Here, we used the fact that 2m < θ.
By using the continuity of h w , we deduce the existence of a global maximum (a, b) ∈ [0, +∞) × [0, +∞) of h w .
Let prove that a, b > 0. Suppose by contradiction that b = 0. Then J ′ (aw + ), aw + = 0. In light of Lemma 2.4, we have Hence, using Lemma 2.3 and taking into account that R N G(aw + ) dx > 0 (see Lemma 4.5-(ii) ), we infer that (4.42) Suppose that a > 1, then On the other hand, since J ′ (w), w + = 0, we have From Lemma 4.2 (with t = s = 1) and taking into account that which is equivalent to (4.45) Putting together (4.43) and (4.45) and using assumption (f 4 ) and Lemma 2.8, we find that Thus the contradiction, so a ≤ 1.
Using Lemma 4.4 and having in mind that w ∈ M, we deduce that Since (a, b) is a global maximum point of h w , a contradiction holds, so b > 0. Similarly, we show that a > 0.
Let prove that a ≤ 1 and b ≤ 1.
Without lose of generality, we can suppose that a ≥ b. The mapping t → g(t)t is increasing and even, then By Lemma 2.4, we get From (4.48) and Lemma 2.3, we conclude that Suppose that a > 1, then On the other side, since J ′ (w), w + = 0, we have which is equivalent to Putting together (4.49) and (4.50) and using (f 4 ) and Lemma 2.8, we obtain 0 ≤ 1 Thus the contradiction, then 0 < b, a ≤ 1. on the other side, in light of Lemma 4.4, we have Thus the proof of (i).
On the other side, we have Using the assumption (g 3 ) and having in mind J ′ (w), w ± = 0 (w ∈ M), we deduce that Thus The proof of (ii) is completed. Now we are able to prove our main result.

Proof of Theorem 2.9
We will prove the existence of w ∈ M such that J(w) = inf v∈M J(v) = c 0 . By using a quantitative deformation lemma, we show that w is a critical point of J, which is a sign-changing solution of (P ).
Consequently there exists (t,s) ∈ D such thatη τ 0 (t,s) ∈ M, thus the contradiction with (4.64). We conclude that w is a critical point of J.