Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent

This paper is dedicated to studying the Choquard equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = (I_{\alpha}\ast|u|^{p})|u|^{p-2}u+g(u),\; \; \; \; \; x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}) ,\\ \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ N\geq4 $\end{document} , \begin{document}$ \alpha\in(0, N) $\end{document} , \begin{document}$ V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R}) $\end{document} is sign-changing and periodic, \begin{document}$ I_{\alpha} $\end{document} is the Riesz potential, \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} and \begin{document}$ g\in\mathcal{C}(\mathbb{R}, \mathbb{R}) $\end{document} . The equation is strongly indefinite, i.e., the operator \begin{document}$ -\Delta+V $\end{document} has infinite-dimensional negative and positive spaces. Moreover, the exponent \begin{document}$ p = \frac{N+\alpha}{N-2} $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on \begin{document}$ g $\end{document} , we obtain the existence of nontrivial solutions for this equation.

(4) Define an inner product and the corresponding norm where (·, ·) L2 denotes the inner product of L 2 (R N ), |·| s denotes the norm of L s (R N ). By (V), it is easy to see that E = H 1 (R N ). Therefore, E is embedded continuously into L s (R N ) for all 2 ≤ s ≤ 2 * . In addition, one has the orthogonal decomposition E = E − ⊕ E + with respect to both (·, ·) L 2 and (·, ·), see [28]. Let us define where 2 * α = 2(N +α) N −2 . It is obvious that finding weak solutions of problem (1) is equivalent to finding critical points of the energy functional Furthermore, Φ is of class C 1 (E, R) and (9) In view of (4) and (6), we have and (11) Problem (1) can be viewed as a local nonlinear perturbation of The case N = 3, p = 2 and α = 2 goes back to the description of the quantum theory of a polaron at rest by Pekar [24] in 1954. The existence of solutions for (12) was proved via variational methods [4,5,6,13,14,20,29,30] and ordinary differential equations techniques [7,21], respectively. There are not a few approaches devoted to the existence and multiplicity of solutions of (12) and their qualitative properties, see, for example, the survey paper [23] and the references therein.
To study problem (12) variationally, the well-known Hardy-Littlewood-Sobolev inequality is the starting point. Usually, N +α N −2 (or N +α N ) is called the upper (or lower) critical exponent with respect to the Hardy-Littlewood-Sobolev inequality (see [15]). For the subcritical autonomous case p ∈ ( N +α N , N +α N −2 ) and V (x) ≡ 1, Moroz and Van Schaftingen [22], as well as established the regularity, positivity and the decay estimates of groundstates, they obtained some nonexistence results under the range p ≥ N +α N −2 or p ≤ N +α N . Thus, for the upper critical case, the subcritical perturbation is necessary to secure the existence of a nontrivial solution.
Recently, there are a few papers study the nontrivial solutions for problem (1). In [31], Van Schaftingen and Xia considered the lower critical problem Under some assumptions on f (u), they obtained a groundstate of (13) by using the Mountain-Pass lemma, the Brezis-Lieb lemma and the concentration compactness principle. Li and Tang [17] considered the upper critical problem with g subcritical and super-quadratic; they obtained a groundstate of (14) by using compactness lemma of Struwe [25]. Recently, Li, Ma and Zhang [18] considered the subcritical autonomous Choquard equation They studied the existence of the nontrivial solutions of (15) under λ > 0, N +α N < p < N +α N −2 and q ∈ (2, 2 * ). In [1], Ackermann discussed the strongly indefinite Choquard-Pekar equation By using the generalized linking theorem of Kryszewski-Szulkin and Bartsch-Ding, the author obtained infinitely many geometrically distinct weak solutions of (16). Chabrowski and Szulkin [3] investigated the semilinear Schrödinger equation with indefinite linear part They estimated the function value in a suitable interval and obtained nontrivial solutions for (17) by means of the linear operator theory. For more results about Schrödinger equation, see [26] and references therein. However, to our knowledge, there seem to be no results on nontrivial solutions for (1) with strongly indefinite linear part and upper critical exponent. Motivated by [3,26] and aforementioned works, in the present paper, we deal with the case when 0 lies in a gap of the spectrum σ(A) and p = N +α N −2 . In addition to (G 1 )-(G 2 ), we also assume that (G 3 ) G(s) := 1 2 g(s)s − G(s) > 0 if s = 0, and there exist c 0 > 0, σ ∈ (0, 1) and r 0 > 0 such that Remark 1. (G 3 ) was first introduced in [26], which is similar to the following assumption (DL) given by Ding and Lee [8]: (DL) F(x, t) > 0 if t = 0, and there exist c 0 > 0, r 0 > 0 and κ > max{1, N/2} such that |f (x, t)| κ ≤ c 0 F(x, t)|t| κ for |t| ≥ r 0 .
However, they are different conditions (see the following example).
Then problem (1) has a nontrivial solution.
To check our result, we must overcome two difficulties in verifying the boundedness and the non-vanishing of the Cerami sequence {u n }. Firstly, since the equation involves the strongly indefinite linear part, it is difficult to show that 1 We will overcome this difficulty by estimating the norm of Ψ 1 (u n ) (see (23)), similar to that of Ackermann [1]. Secondly, since the equation with critical growth brings many obstacles for recovering the compactness, it becomes very hard to prove that the Cerami sequence {u n } is non-vanishing; we can't apply the arguments in [3] directly because of the absorption of the convolution term. We shall introduce some new tricks to overcome this difficulty. The paper is organized as follows. We present some preliminary results in section 2; give our variational framework in section 3; give the proof of Theorem 1.1 in section 4.
2. Preliminary lemmas. Let X be a real Hilbert space with X = X − ⊕ X + and X − ⊥ X + . For a functional ϕ ∈ C 1 (X, R), ϕ is said to be weakly sequentially lower semi-continuous if for any u n u in X one has ϕ(u) ≤ lim inf n→∞ ϕ(u n ), and ϕ is said to be weakly sequentially continuous if Lemma 2.1 ( [12,16]). Let (X, · ) be a real Hilbert space, with X = X − ⊕ X + and X − ⊥ X + , and let ϕ ∈ C 1 (X, R) with the form Suppose that the following assumptions are satisfied: (KS1) ψ ∈ C 1 (X, R) is bounded from below and weakly sequentially lower semicontinuous; (KS2) ψ is weakly sequentially continuous; (KS3) there exist r > ρ > 0 and e ∈ X + with e = 1 such that Then there exist a constant c ∈ [κ, sup ϕ(Q)] and a sequence {u n } ⊂ X satisfying The following well-known Hardy-Littlewood-Sobolev inequality which can be found in [15] will be repeatedly used in this paper.
Lemma 2.2. Let 0 < α < N , and s, r > 1 be constants such that Assume that f ∈ L s (R N ). Then there exists a sharp constant C(N, α, s), independent of f , such that As in the proof of [31, Proposition 2.4], we have the following lemma.
The following quasi-Cauchy-Schwarz inequality is the key to estimating the norm of Ψ 1 (u n ).

Lemma 2.4 ([19]
). Let N ∈ N, · 1 and · 2 be any two norms on R N . Then there is a constant In order to prove that the Cerami sequence is non-vanishing, we need the following lemma.
Proof. Let u n u in E, up to a subsequence, u n → u, a.e. on R N . Then, by Fatou's lemma, we have Next, we show that ψ is weakly sequentially continuous. Assume that u n u in E. By the Sobolev embedding theorem, u n is bounded in L 2 * (R N ). Therefore, the Using Lemma 2.2, we have For any R > 0, by (19), there holds Therefore, for every ε > 0 and every R > 0, Then, by Riesz Lemma, up to a subsequence, one gets a.e. on R N . Hence, Therefore, for any v ∈ E, On the other hand, it follows from (G 2 ) that Taking R large enough, one has The proof is thus finished.
The following lemma is crucial to the verification of the link geometry of Φ.
Use S α to denote the best constant defined by The constant S α is achieved by where C > 0 is a fixed constant and ε > 0 is a parameter (see [11,15]). Let ψ ∈ C ∞ 0 (R N , [0, 1]) be a cut-off function satisfying ψ = 1 for x ∈ B ρ and ψ = 0 for We shall need the following asymptotic estimates as ε → 0 + (see [2,11,32]): and Now we are ready to estimate the "Cerami level" c given in Lemma 2.1.