Existence of nontrivial weak solutions for a quasilinear Choquard equation

We are concerned with the following quasilinear Choquard equation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\Delta_{p} u+V(x)|u|^{p-2}u=\lambda\bigl(I_{\alpha} \ast F(u)\bigr)f(u) \quad \text{in } \mathbb {R}^{N}, \qquad F(t)= \int_{0}^{t}f(s) \,ds, $$\end{document}−Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫0tf(s)ds, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p<\infty$\end{document}1<p<∞, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta_{p} u=\nabla\cdot(|\nabla u|^{p-2}\nabla u)$\end{document}Δpu=∇⋅(|∇u|p−2∇u) is the p-Laplacian operator, the potential function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V:\mathbb {R}^{N}\to(0,\infty)$\end{document}V:RN→(0,∞) is continuous and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F \in C^{1}(\mathbb {R}, \mathbb {R})$\end{document}F∈C1(R,R). Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\alpha}: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$\end{document}Iα:RN→R is the Riesz potential of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\in(0,p)$\end{document}α∈(0,p). We study the existence of weak solutions for the problem above via the mountain pass theorem and the fountain theorem. Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f.


Introduction
We are concerned with the following quasilinear Choquard equation: where 1 < p < N , p u = ∇ · (|∇u| p-2 ∇u) is the p-Laplacian operator, the potential function V : R N → (0, ∞) is continuous and F ∈ C 1 (R, R) with F(t) = t 0 f (s) ds. Here, I α : R N → R is the Riesz potential of order α ∈ (0, p) on the Euclidean space R N of dimension N ≥ 1, defined for each x ∈ R N \ {0} by where (·) stands for a standard Gamma function. The Choquard equation was also introduced by Choquard in 1976 in the modeling of a one-component plasma [1]. It seems to originate from the Frohlich and Pekarí's model of the polaron, which is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material [2,3]. This equation is also known as the Schrödinger-Newton equation in models coupling the Schrödinger equation of quantum physics together with relativistic or nonrelativistic Newtonian gravity [4,5]. Thus, they have become very significant in physics (see [6] for a review paper). For this reason, many researchers have extensively studied the Choquard type equation in various ways; see [7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.
Recently, the authors [19] dealt with the existence of positive solutions to the problem (P) on the whole space R N , based on the assumption that the nonlinearity f satisfies the following Ambrosetti-Rabinowitz superlinear condition [20], which is commonly called the (AR)-condition: 0 < ζ F(t) ≤ θ f (t)t for t > 0 and some θ ∈ 0, 1 2 , where F(t) = t 0 f (s) ds. It is well known that the (AR)-condition is quite natural and important not only to ensure that an Euler-Lagrangian functional has the mountain pass geometry, but also to guarantee that the Palais-Smale sequences of the functional are bounded. However, this condition is very restrictive and eliminates many nonlinearities. Thus, many researchers have tried to drop the (AR)-condition for elliptic equations involving the p-Laplacian; see e.g. [21][22][23][24].
The purpose of this paper is to study the existence of weak solutions for the problem (P) without the (AR)-condition as observing various assumptions for the nonlinear term f compare to result in [19]. In particular, following Remark 1.8 in [21], there are many examples which do not fulfill the condition of f given in [22,23,25]. On the other hand, in the case of the whole space R N , the main difficulty of this problem is the lack of compactness for the Sobolev theorem and we introduce the potential V to the equation. To be precise, we prove the existence of weak solutions for the quasilinear Choquard equation (P) under the Cerami condition, as a weak version of the Palais-Smale condition. To do this, first, we use the uniform boundedness of the convolution part, |I α * F| < ∞ for our analysis (see Section 3.1 for a detailed description) and thus the property of (S + ) type operator with this uniform estimate gives a lot of help when we choose Cerami sequences. Second, we show the multiplicity of weak solutions to the quasilinear Choquard equation (P) via the fountain theorem to obtain the infinitely many weak solutions. Third, we establish the existence of a sequence of weak solutions for the problem (P) converging to zero to obtain the L ∞ -bound of weak solutions to the problem (P) based on an iteration method. To the best of our knowledge, there were no such existence results for our problem in this situation.

Preliminaries
Let 1 < p < N and p * := Np/(Np) denote the Sobolev conjugate of p. Suppose that Then X is a reflexive separable Banach space with the norm which is equivalent to the norm · W 1,p (R N ) given by We recall the well-known embedding results in [21, Lemma 2.1]; see also [26].

Lemma 2.1 The following statements hold:
(i) There is a continuous embedding W 1,p Throughout this paper, let X be the completion of C ∞ 0 (R N , R), and X * be a dual space of X. Furthermore, ·, · denotes the pairing of X and its dual X * . All generic constants will be denoted by C, which may vary from line to line.

Definition 2.2
We say that u ∈ X is a weak solution of the problem (P) if Let us define the functional : X → R by Under the assumption (V), it is obvious that the functional is well defined on X, ∈ C 1 (X, R) and its Fréchet derivative is given by We suppose that the following assumptions hold: (F3) There exists δ > 0 such that |t| p = ∞. (F5) There exist c 0 ≥ 0, r 0 ≥ 0, and κ > N p such that for all t ∈ R and |t| ≥ r 0 , where F(t) : To comment on the assumptions about the nonlinearity f , we would like to recall an important inequality due to [1].
Let s, r > 1 and α < p with 1/s + (Nα)/N + 1/r = 2. Let g ∈ L s (R N ) and h ∈ L r (R N ). Then there exists a sharp constant C(s, N, α, r), independent of g and h, such that In particular, F(t) = |t| q 1 for some q 1 > 0. By the Hardy-Littlewood-Sobolev inequality, Since u ∈ W 1,p (R N ), we must require that sq 1 ∈ [p, p * ]. For the subcritical case, we must assume Next we define the functional : X → R by Then it is easy to check that ∈ C 1 (X, R) and its Fréchet derivative is for any u, v ∈ X. Also we define the functional I λ : X → R by Then it follows that the functional I λ ∈ C 1 (X, R) and its Fréchet derivative is According to similar arguments in [27, Theorem 4.1], the following lemma is easily checked, and thus we omit the proof. That is, the operator is of type (S + ); see [28].

Lemma 2.3
Assume that the assumption (V) holds. Then the functional : X → R is convex and weakly lower semicontinuous on X. Moreover, the operator is of type (S + ), i.e., if u n u in X and lim sup n→∞ (u n )-(u), u n -u ≤ 0, then u n → u in X as n → ∞.
In our setting, we need the following lemma according to a similar argument in [29, Lemma 3.2]. We give a detailed proof for the convenience of the reader.

Lemma 2.4
Assume that (V) and (F1)-(F2) hold. Then and are weakly strongly continuous on X.
Proof See the Appendix.

Existence of weak solutions
In this section, we shall give the proof of the existence of nontrivial weak solutions for the problem (P), by applying the mountain pass theorem and the fountain theorem.
With the aid of Lemmas 2.3 and 2.4, we prove that the energy functional I λ satisfies the Cerami condition ((C) c -condition for short), i.e., for c ∈ R, any sequence {u n } ⊂ X such that has a convergent subsequence. This plays a key role in obtaining the existence of a nontrivial weak solution for the given problem. Before investigating a crucial lemma, we note that following [19], there exists M > 0, such that (3.1) Indeed, by the assumption (F2), Hence this inequality implies the uniform boundedness (3.1) for the convolution part.

Existence of weak solutions: approach to the mountain pass theorem
We give the following result to show that the energy functional I λ satisfies the geometric conditions of the mountain pass theorem based on the idea of Lemma 3.2 in [30].
Proof In view of the assumption (F3), u = 0 is a strict local minimum for I λ (u). Next we show that the condition (2) holds. By the assumption (F4), we can take s 0 such that F(s 0 ) = 0 and we have to find where B r denotes the open ball centered at the origin with radius r and 1 A denotes the standard indicator function of a set A. Due to the density theorem, there will be v 0 ∈ X with . This function verifies for sufficiently large t. Therefore, we assert that I λ (v t ) → -∞ as t → ∞. Hence we conclude that the functional I λ is unbounded from below. This completes the proof. Proof For c ∈ R, let {u n } be a (C) c -sequence in X, that is, This says that where o(1) → 0 as n → ∞. It follows from Lemmas 2.3 and 2.4 that I λ is of type (S + ). Since I λ is of type (S + ) and X is reflexive, it suffices to prove that the sequence {u n } is bounded in X. We argue by contradiction. Suppose that the sequence {u n } is unbounded in X. Then we may assume that u n X > 1 and u n X → ∞ as n → ∞. Define a sequence {w n } by w n = u n / u n X . It is clear that {w n } ⊂ X and w n X = 1. Hence, up to a subsequence, still denoted by {w n }, we obtain w n w in X as n → ∞ and note that for p ≤ s < p * . According to (3.1), we obtain Since u n X → ∞ as n → ∞, we have In addition, it follows from Eq. (3.2) that for sufficiently large n. The assumption (F4) implies that there exists t 0 > 1 such that F(t) > |t| p for all |t| > t 0 . From the assumptions (F1) and (F2), there exists C > 0 such that |F(t)| ≤ C for all t ∈ [-t 0 , t 0 ]. Therefore we can choose a real number C 0 such that F(t) ≥ C 0 for all t ∈ R, and thus for all n ∈ N. Therefore there exists a real number C 1 such that for all x ∈ 1 . So then it follows from the assumption (F4) and Hölder's inequality that, for all x ∈ 1 , we have Hence we get meas( 1 ) = 0. Indeed, if meas( 1 ) = 0, according to (3.5)-(3.6) and Fatou's lemma, we obtain for p ≤ r < p * . Hence by the relation (3.4) we get On the other hand, from the assumption (F2) and Eq.
Using Lemma 3.2, we prove the existence of a nontrivial weak solution for our problem under the assumptions. Proof Note that I λ (0) = 0. In view of Lemma 3.1, the geometric conditions in the mountain pass theorem are fulfilled. And also, I λ satisfies the (C) c -condition for any λ > 0 by Lemma 3.2. Hence, the problem (P) has a nontrivial weak solution for all λ > 0. This completes the proof.

Existence of a sequence of weak solutions: approach to the fountain theorem
In this subsection, applying the fountain theorem in [31,Theorem 3.6] with the oddity on f , we investigate infinitely many weak solutions for the problem (P). For this purpose, let W be a reflexive and separable Banach space. Then there are {e n } ⊆ W and {f * n } ⊆ W * such that W = span{e n : n = 1, 2, . . .}, W * = span f * n : n = 1, 2, . . . , Let us denote W n = span{e n }, Y k = k n=1 W n , and Z k = ∞ n=k W n .

Lemma 3.4
Let X be a real reflexive Banach space. Suppose that I ∈ C 1 (X, R) satisfies the (C) c -condition for any c > 0 and I is even. If for each sufficiently large k ∈ N, there exist ρ k > δ k > 0 such that the following conditions hold: (2) a k := max{I(u) : u ∈ Y k , u X = ρ k } ≤ 0. Then the functional I has an unbounded sequence of critical values, i.e., there exists a sequence {u n } ⊂ X such that I (u n ) = 0 and I(u n ) → ∞ as n → ∞. Proof It is obvious that I λ is an even functional and satisfies the (C) c -condition. It suffices to show that there exist ρ k > δ k > 0 such that (1) b k := inf{I λ (u) : u ∈ Z k , u X = δ k } → ∞ as k → ∞; (2) a k := max{I λ (u) : u ∈ Y k , u X = ρ k } ≤ 0 for k large enough. Denote Then we have α k → 0 as k → ∞. In fact, assume to the contrary that there exist ε 0 > 0, k 0 ∈ N, and a sequence {u k } in Z k such that u k X = 1 and u k L q 1 (R N ) + u k L q 2 (R N ) ≥ ε 0 for all k ≥ k 0 . By the boundedness of the sequence {u k } in X, we can find an element u ∈ X such that u k u in X as k → ∞ and for j = 1, 2, . . . . Thus we deduce u = 0. However, we see that which is a contradiction. For any u ∈ Z k , we may suppose that u X > 1. According to the assumption (F2), we obtain where q i is either q 1 or q 2 . If we take , then δ k → ∞ as k → ∞ because p < q i and α k → 0 as k → ∞. Hence, if u ∈ Z k and u X = δ k , then we conclude that This implies that the condition (1) holds. The proof of the condition (2) proceeds analogously as in the proof of Theorem 1.3 of [25]. For the reader's convenience, we give the proof. Assume that the condition (2) is not true. Then, for some k there exists a sequence {u n } in Y k such that u n X → ∞ as n → ∞ and I λ (u n ) ≥ 0. (3.12) Set w n = u n / u n X . Note that w n X = 1. Since dim Y k < ∞, there exists w ∈ Y k \ {0} such that up to a subsequence, for almost all x ∈ R N as n → ∞. If w(x) = 0, then |u n (x)| → ∞ for all x ∈ R N as n → ∞.
Hence we obtain by the assumption (F4) that for all x ∈ 2 := {x ∈ R N : w(x) = 0}. A similar argument to (3.6) proves that 2 (I α * F(u n ))F(u n ) u n p X dx → ∞ as n → ∞.
Therefore, we conclude that which contradicts (3.12). This completes the proof.

Existence of a sequence of weak solutions converging to zero
Now, we deal with the existence of a sequence of weak solutions converging to zero for the problem (P). First of all, we need the following additional assumptions for f : |t| p-2 t = +∞. From the assumptions above, we show the existence of a sequence of solutions for the problem (P) converging to zero in the L ∞ -norm based on the iteration method in [32,Theorem 4.1]. Since the problem (P) contains the potential term V , more sophisticated analysis has to be carefully carried out in comparison to the result in [32] (compare to [33] for the bounded domain). Proof The proof is given in the Appendix.
The following lemma is quoted from [34].

Lemma 3.7
Let I ∈ C 1 (X, R) where X is a Banach space. Assume I satisfies the (PS)condition, is even and bounded from below, and I(0) = 0. If for any n ∈ N, there exist an n-dimensional subspace X n and ρ n > 0 such that sup X n ∩S ρn where S ρ := {u ∈ X : u X = ρ}, then I has a sequence of critical values c n < 0 satisfying c n → 0 as n → ∞.
Now, for convenience of the reader, we prove the following result using Proposition 3.6 and Lemmas 3.7 and 3.9 (see e.g. [35, pp. 18-21]). Proof First of all, we claim that I λ is coercive on X. Let u ∈ X and u X > 1. For the given function f , we can modify and extendf ∈ C 1 (R, R) satisfying all properties listed in Lemma 3.9 with ξ such that λpCξ ≤ 1. And also by Lemma 3.9, it is easy to show that I λ ∈ C 1 (X, R) and is even on X. Moreover, it follows from (F2) that, for |u(x)| ≤ 2t 0 for a sufficiently small t 0 , there exists a positive constant M 1 such that |F(u)| ≤ M 1 |u| p . Set 3 := {x ∈ R N : |u(x)| ≤ t 0 }, 4 := {x ∈ R N : t 0 ≤ |u(x)| ≤ 2t 0 }, and 5 := {x ∈ R N : 2t 0 ≤ |u(x)|}, where t 0 is given in Lemma 3.9. From (F8), (3.13), and the conditions of κ, we havẽ If we set then for every λ ∈ [0, λ * ) we haveĨ λ is coercive, that is,Ĩ λ (u) → ∞ as u X → ∞. By a standard argument,Ĩ λ satisfies the (PS) c -condition. In order to apply Lemma 3.7, we only need to find for any n ∈ N, a subspace X n and ρ n > 0 such that sup X n ∩S ρnĨ λ < 0. For any n ∈ N we find n independent smooth functions φ i for i = 1, . . . , n, and define X n := span{φ 1 , . . . , φ n }. Due to Lemma 3.9, when u X < 1 we havẽ (3.14) It follows from the assumption (F8) that, for a sufficiently large M 2 > 0, there exists δ 0 > 0 such that |t| < δ 0 implies F(t) ≥ M 2 p |t| p and Combining this and the fact that all norms on X n are equivalent, choosing a suitable constant C and sufficiently small ρ n > 0, we can obtain by (3.14) that sup X n ∩S ρnĨ λ < 0.
By Lemma 3.7, we get a sequence c n < 0 forĨ λ satisfying c n → 0 when n goes to ∞. Then, for any u n ∈ X satisfyingĨ λ (u n ) = c n andĨ λ (u n ) = 0, the sequence {u n } is a (PS) 0 -sequence ofĨ λ (u) and {u n } has a convergent subsequence. Lemmas 3.8 and 3.9 imply that 0 is the only critical point with zero energy and the subsequence of {u n } has to converge to 0. An indirect argument shows the sequence {u n } has to converge to 0. On the other hand, we have u n ∈ C(R N , R) due to Proposition 3.6. Since u n L ∞ (R N ) → 0, by Lemma 3.9 again, we deduce u n C(R N ) ≤ t 0 . Thus {u n } are weak solutions of the problem (P). The proof is complete.

Conclusion
In this paper, we obtain the existence of nontrivial weak solutions for a quasilinear Choquard equation on the whole space R N without (AR)-condition based on the uniform boundedness of the convolution part in the Choquard term driven by the Riesz potential.
Moreover, the existence of infinitely many weak solutions is obtained via the fountain theorem. Lastly, we prove that our problem has a sequence of solutions converging to zero in the L ∞ -norm based on the iteration method.
Our arguments also allow one to prove Theorems 3.3 and 3.5 for the p(x)-Laplacian equation the potential V satisfying the assumption (V). Define the linear subspace which is equivalent to the following norm: Under this circumstance, we introduce the functional J λ : X → R by Then it follows that the functional J λ ∈ C 1 (X, R) and its Fréchet derivative is for any u, v ∈ X. In order to show the boundedness of the Cerami sequence, we use the boundedness of the convolution part (3.1). For some properties of the variable exponent Sobolev space, we refer to [25]. And hence we omit the details proof.
Then we may suppose that u n → u in L q 1 (R N ) ∩ L q 2 (R N ) as n → ∞. By the convergence principle, there exist a subsequence {u n k } such that u n k (x) → u(x) as k → ∞ for almost all x ∈ R N and a function u 0 ∈ L q 1 (R N ) ∩ L q 2 (R N ) such that |u n k (x)| ≤ u 0 (x) for all k ∈ N and for almost all x ∈ R N . First we prove that is weakly strongly continuous on X. Since F ∈ C 1 (R, R), we see that F(u n k ) → F(u) as k → ∞ for almost all x ∈ R N and so (I α * F(u n k ))F(u n k ) → (I α * F(u))F(u) as k → ∞. From (F2), it follows that Therefore, the Lebesgue convergence theorem tells us that as k → ∞, which implies (u n k ) → (u) as k → ∞. Thus is weakly strongly continuous on X.
Next, we show that is weakly strongly continuous on X.
By (F2) and Hölder's inequality, we obtain for any ϕ ∈ X Combining this with the Lebesgue convergence theorem, we have Therefore, we derive that (u n k ) → (u) in X * as k → ∞. This completes the proof.

A.2 Proof of Proposition 3.6
Following [36], we give the proof of Proposition 3.6. (k ≥ 0) as a test function in (2.1). Then obviously v ∈ X ∩ L ∞ (R N ) and it follows from (2.1) that

Proof
Due to Lemma 2.1, the left-hand side of (A.1) can be estimated as follows: for some constant C 1 > 0. By using the assumption (F2) and the Hölder inequality, the right-hand side of (A.1) can be formally estimated from above and we obtain where s i = pp * p * -(q i -p) for i = 1, 2. Obviously p < s i ≤ p * and (q i +1-p)s i s i -p = p * for i = 1, 2, and hence (A.3) yields Since (k n + 1) 1 kn+1 > 1 and lim k n →∞ (k n + 1) 1 kn+1 = 1, there exists C 4 > 1 (independent of k n ) such that u L (kn+1)p * (R N ) ≤ C Then it is clear that u + ∈ X and u -∈ X. Proceeding by a similar argument to above, we obtain u + ∈ L ∞ (R N ). Likewise, we get u -∈ L ∞ (R N ). Therefore u = u + + uis in L ∞ (R N ). This completes the proof.