Generalized Choquard Equations Driven by Nonhomogeneous Operators

In this work we prove the existence of solutions for a class of generalized Choquard equations involving the ΔΦ-Laplacian operator. Our arguments are essentially based on variational methods. One of the main difficulties in this approach is to use the Hardy–Littlewood– Sobolev inequality for nonlinearities involving N-functions. The methods developed in this paper can be extended to wide classes of nonlinear problems driven by nonhomogeneous operators. Mathematics Subject Classification. 35A15, 35J62, 35J60.


Introduction
The stationary Choquard equation where N ≥ 3, 0 < λ < N, has appeared in the context of various physical models. In particular, this equation plays particularly an important role in the theory of Bose-Einstein condensation where it accounts for the finite-range many-body interactions. For N = 3, p = 2, and λ = 1, problem (1.1) was investigated by Fröhlich [1] and Pekar [2] in relationship with the quantum theory of a polaron, where free electrons in an ionic lattice interact with phonons associated to deformations of the lattice or with the polarisation that it creates on the medium (interaction of an electron with its own hole). We recall that Choquard [3] used this equation in the Hartree-Fock theory of one-component plasma. This equation was also proposed by Penrose in [4] as a model of self-gravitating matter and is known in that context as the Schrödinger-Newton equation. In fact, the Choquard equation is also known as the Schrödinger-Newton C. O. Alves et al. MJOM equation in models coupling the Schrödinger equation of quantum physics together with nonrelativistic Newtonian gravity. Recent relevant contributions included in the papers are by Ackermann [5], Alves et al. [6][7][8], Cingolani, Secchi and Squassina [9], Gao and Yang [10], Lions [11], Ma and Zhao [12], Moroz and van Schaftingen [13][14][15][16], van Schaftingen and Xia [17], Wang [18], and their references.
We also assume that there exist l, m ∈ (1, N) such that l ≤ m < l * := Nl is well defined and belongs to C 1 (W 1,Φ (R N ), R). In fact, the main difficulty is to prove that the functional Ψ : In what follows, we would like to point out that the operator Δ Φ arises in several physical applications, such as: Non-Newtonian fluids: The reader can find more details about the physical applications in [20], [21] and their references. The existence of solution for with Ω ⊂ R N being a bounded or unbounded domain has been established in some papers, see for example [22][23][24][25][26][27][28][29][30][31][32][33][34][35] and the references therein.
This paper is organized as follows. In Sect. 2 we recall some facts involving Orlicz-Sobolev spaces. In Sect. 3 we show that Ψ is of class C 1 under certain conditions on f. In Sect. 4 we study the existence of solutions to problem (1.3). Finally, in Sect. 5, we show other problems that can be studied with the approach developed in the present paper.

Orlicz-Sobolev Spaces
In this section we recall some results on Orlicz-Sobolev spaces. The results pointed below can be found in [21,28,36,37].
We say that a continuous function Φ : (iv) Φ is even. We say that a N-function Φ verifies the Δ 2 -condition, and we denote by Alves et al. MJOM for some constants K, t 0 > 0. An important fact is that if Φ verifies the Δ 2 -condition then for some s > 0 there is a constant C s > 0 such that We fix an open set Ω ⊂ R N and a N-function Φ. We define the Orlicz space associated with Φ as follows The space L Φ (Ω) is a Banach space endowed with the Luxemburg norm given by In the case of |Ω| = +∞ we will consider that Φ ∈ Δ 2 if t 0 = 0 in (2.1). The complementary function Φ associated with Φ is given by the Legendre transformation, that is, We also have a Young-type inequality given by Using the above inequality, it is possible to establish the following Höldertype inequality: The following results will be often used and they can be found in [21,28].
For a N -function Φ, the corresponding Orlicz-Sobolev space is defined as the Banach space endowed with the norm If Φ and Φ satisfy the Δ 2 -condition, then the spaces L Φ (Ω) and W 1,Φ (R N ) are reflexive and separable. Moreover, the Δ 2 -condition also implies that An important function related to a N -function Φ is the Sobolev conjugate function Φ * of Φ defined by Let Φ 1 and Φ 2 be N -functions. We say that Φ 1 increases strictly lower than Φ 2 , and we denote by We say that Ω ⊂ R N is an admissible domain, if the embedding The following embedding results can be found in [25,38].
The next result can be found in [25] and it will play an important role in this work.
Then, for any N-function P verifying the Δ 2 -condition with we have

Differentiability of the Functional Ψ
In this section, we will study the differentiability of the functional Ψ given in (1.5). To this end, we must assume some conditions on f . We will consider B : R → [0, +∞) being a N -function given by is a function satisfying the following conditions: The above hypotheses permit to use Lemmas 2.1 and 2.2 by changing Φ by B. More precisely, B satisfies the Δ 2 condition with Let f : R → R be a continuous function satisfying the growth condition where C is a positive constant.
The primitive of f , that is, F (t) := t 0 f (s)ds is continuous and satisfies We will also suppose that the embeddings are continuous. The above condition is not empty, because we can consider the following conditions on B lim sup If Φ(t) = 1 p |t| p with 1 < p < N, we have that Δ Φ u = Δ p u and the function f can be of the form In the proof of the differentiability of Ψ, we will use the following elementary property, whose proof we omit. Lemma 3.1. Let E be a normed vector space and J : E → R be a functional verifying the following properties: Then J ∈ C 1 (E, R) and We are now ready to prove the differentiability of functional Ψ given by (1.5).
Proof. Using the definition of s, the condition (F ) and Proposition 1.1, it follows that Ψ is well defined. In the sequel, we will show that Ψ satisfies the assumptions of Lemma 3.1. To this end, we will divide the proof into three steps. C. O. Alves et al. MJOM Step 1: Existence of the Fréchet derivative: Denoting by I the integrand in (3.1), we have By the mean value theorem, there exist θ( The relation (3.1) allows us to estimate The above estimate also holds if u(x) = 0. Such estimate, combined with (E) Thus, by Fubini's theorem, Therefore, B t 2 can be rewritten as By Proposition 1.1 we obtain The embeddings (E) ensure that the right-hand side of the inequality (3.2) is an integrable function. Thus, the Lebesgue's dominated convergence theorem yields Arguing as before, as t → 0. On the other hand, the Lebesgue's dominated convergence theorem also yields as t → 0, and so, B t 1 → 0 as t → 0. From the above analysis, showing the existence of the Fréchet derivative ∂Ψ(u) ∂v . C. O. Alves et al. MJOM Step 2: ∂Ψ(u) ∂(.) ∈ (W 1,Φ (R N )) for all u ∈ W 1,Φ (R N ). It is evident that ∂Ψ(u) ∂v is linear at v for each u fixed. Next, we are going to show that for some positive constant C u . From (f ), (F ) and Proposition 1.1 we have The continuous embeddings of (E) combined with Hölder inequality, (f ) and Proposition 2.1 give where with K a constant that does not depend on u and v. Then inequalities (3.4) and (3.5) justify Step 2.
Step 3: Since the sequence ( f (u n )v L s (R N ) ) is bounded (see (3.5)) and (3.6) Now we will estimate B n F . Given ε > 0, fix R > 0 large enough such that Since u n → u in L b1s (R N ) and L b2s (R N ), there is n 0 ∈ N large enough such that for all n ≥ n 0 . Note that by Proposition 1.1 where C 0 > 0 is a constant that does not depend on n ∈ N. The condition (f ) together with Hölder's inequality yields where C 1 > 0 is a constant that does not depend on n ∈ N. Using the embeddings (E), we have v s L sb i (B(0,R) c ) ≤ C 2 , i = 1, 2 where C 2 is a positive constant that does not depend on v ∈ W 1,Φ (R N ) with v W 1,Φ (R N ) ≤ 1 and R > 0. Thus, from (3.7) and (3.9) where C 3 is a positive constant that does not depend on |f (u n (y)) − f (u(y))| s |v(y)| s dy.

Existence of Nontrivial Solutions
To prove the existence of solution for (1.3) we will need some assumptions.
In what follows, we will consider m < sb i < l * , i = 1, 2, which implies that the embeddings in (E) hold. Moreover, we will consider the condition (f 1 ) with Finally, we also assume the Ambrosetti-Rabinowitz-type condition: there is θ > m such that Our main result is the following: In the proof of Theorem 4.1 we will use variational methods. The energy functional J : W 1,Φ (R N ) → R associated with (1.3) is given by, that is, The analysis developed in the previous section implies that J ∈ Our first lemma establishes the mountain pass geometry. Proof. (i) By (F ) and Proposition 1.1 we have From (E) and Theorem 2.1, it follows that the embeddings W 1,Φ (R N ) → L sbi (R N ), i = 1, 2 are continuous. Therefore Alves et al. MJOM By using the classical inequality and Lemma 2.1, we get for where C, C, and K are constants that do not depend on u. Since (4.1) holds, the result follows by fixing u 1,Φ = ρ with ρ > 0 small enough.
(ii) The condition (f 2 ) implies that where C 1 , C 2 ≥ 0 depends only on l and θ. Now, considering a nonnegative function ϕ ∈ C ∞ 0 (R N )\{0}, the last inequality permits to conclude that J(tϕ) < 0 for t large enough. This finishes the proof.
Using the mountain pass theorem without the Palais-Smale condition (see [39,Theorem 5.4 Regarding the above sequence we have the following auxiliary property.
Proof. Note that for n large enough and d given in (4.2). On the other hand by (φ 3 ) and (f 2 ) we have The last two inequalities give the boundedness of (u n ) in W 1,Φ (R N ).
Since (u n ) is bounded in W 1,Φ (R N ), there exists u ∈ W 1,Φ (R N ) such that ∇u n ∇u in (L Φ (R N )) N up to a subsequence. From (4.1) and Theorem 2.1, we have that the embedding W 1,Φ (B R (0)) → L Φ (B R (0)), where B R (0) denotes the open ball centered at the origin with radius R is compact.
Since L Φ (B R (0)) → L 1 (B R (0)), it follows that u n (x) → u(x) a.e in B R (0) for some subsequence. Since R > 0 is arbitrary, we have that u n (x) → u(x) a.e in R N for some subsequence.
The next two lemmas will be needed to prove that u is a critical point of J.

Lemma 4.3. The following limits hold for a subsequence:
(i) The hypothesis (F ), the fact that (u n ) is bounded in W 1,Φ (R N ) and the continuous embeddings W 1,Φ (R N ) → L sbi (R N ), i = 1, 2 ensure that (F (u n (.))) is a bounded sequence in L s (R N ). Combining the previous information with the pointwise convergence F (u n (x)) → F (u(x)) a.e in R N , we have F (u n (.)) F (u) in L s (R N ). By Proposition 1.1 it follows that the function defines a continuous linear functional. Since F (., u n ) F (., u) in L s (R N ), it follows that which proves (i). C. O. Alves et al. MJOM (ii) Denote by I the integral described in (ii). Since (F (u n (.))) is bounded in L s (R N ), for some positive constant K > 0 that does not depend on n ∈ N and v ∈ C ∞ 0 (R N ).
Let v ∈ C ∞ 0 (R N ) and consider a bounded open set Ω that contains the support of v. Since Ω is bounded the compactness of the embeddings W 1,Φ (Ω) → L sbi (Ω), i = 1, 2 implies that there exist c ∈ L sb1 (Ω) and d ∈ L sb2 (Ω) such that These information combined with Lebesgue's dominated convergence theorem give This finishes the proof of (ii). (iii) is a direct consequence of (i) and (ii).
The next lemma is crucial to prove that u is a critical point of J.

Lemma 4.4. There is a subsequence such that
Proof. (i) We begin this proof observing that (φ 1 ) yields where (·, ·) denotes the usual inner product in R N . Given R > 0, let us |x − y| λ dxdy. (4.5) The boundedness of (u n ) in W 1,Φ (R N ) implies that where C is a constant that does not depend on (u n ). Since it follows from (4.6) that the sequence (φ(|u n |)|u n |) is bounded in L Φ (B 2R (0)). A similar reasoning implies that (φ(|∇u n |)|∇u n |) is bounded in L Φ (B 2R (0)).
The compact embedding W 1,Φ (B 2R (0)) → L Φ (B 2R (0)) implies that u n −u L Φ (B2R(0)) → 0 for a subsequence. It follows that, up to a subsequence, we have A similar reasoning implies that for some subsequence. Standard arguments imply, for a subsequence, that Note that C. O. Alves et al. MJOM Thus from Hölder inequality, it follows that the function with v ∈ L Φ (B 2R (0)) defines a linear continuous functional. Since From (4.4), (4.5), (4.7), (4.8) (4.9), (4.10) we obtain that Applying a result found in Dal Maso and Murat [40], it follows that ∇u n (x) → ∇u(x) a.e in B R (0), for each R > 0. Since R is arbitrary, there is a subsequence of (u n ), still denoted by itself, such that Then the boundedness of (u n ) in W 1,Φ (R N ) implies that the sequences φ(|∇u n |) ∂u n ∂x i , i = 1, . . . , N are bounded in L Φ (R N ). By [41], it follows that φ(|∇u n |) A similar reasoning implies (iii). Now, we are ready to prove that u is a critical point of J. Proof. First of all, we claim that . To verify such limit, note that By Lemmas 4.3 and 4.4, 12) and From the relations (4.11), (4.12) and (4.13) we have the claim. Since J (u n )v → 0, the claim ensures that J (u)v = 0, for all v ∈ C ∞ 0 (R N ). Now, the lemma follows using the fact that C ∞ 0 (R N ) is dense in W 1,Φ (R N ).

Proof of Theorem 4.1
If u = 0, then u is a nontrivial solution and the theorem is proved. If u = 0, we must find another solution v ∈ W 1,Φ (R N ) \ {0} for the equation (1.3). For such purpose, the claim below is crucial in our argument. Proof. In fact, if the above claim does not hold, using Theorem 2.2, we derive the limit R N P (|u n |) dx → 0, (4.14) for any N -function P satisfying (P 1 ) − (P 2 ). Applying Proposition 1.1, |f (u n (y))u n (y)| s dy → 0. Therefore R N F (u n (x))f (u n (y))u n (y) |x − y| λ dxdy → 0.
This limit leads to J(u n ) → 0, which contradicts the limit J(u n ) → d > 0.
Using standard arguments, we can assume in Claim 4.1 that (y n ) ⊂ Z N . By setting v n (x) = u n (x + y n ), it follows that J(v n ) = J(u n ), J (v n ) = J (u n ) and u n 1,Φ = v n 1,Φ ∀n ∈ N.
From the above information, we have that J(v n ) → d and J (v n ) → 0. Since (v n ) is bounded in W 1,Φ (R N ), up to a subsequence, v n → v in L Φ (B r (0)), for some v ∈ W 1,Φ . To verify that v = 0, note that by Claim 4.1, we have for some subsequence 0 < β ≤ lim Applying the same arguments as in the proofs of Lemmas 4.3, 4.4 and 4.5 for the sequence (v n ) we obtain the desired result.

Final Comments
The same arguments used in this paper can be applied to study the existence of solutions for related problems of the following type: |x − y| λ f (u(y)), in R N , u ∈ W 1,Φ (R N ). (5.1) The potential V : R N → R is a continuous functions with inf x∈R N V (x) > 0 that belongs to one of the following classes: Class 1: V is periodic: V is a Z N -periodic function, that is, Class 2: V is asymptotic periodic function: There is a Z N -periodic function V p : R N → R such that V (x) < V p (x), ∀x ∈ R N and |V (x) − V p (x)| → 0 as |x| → +∞.