Existence of ground state solutions for quasilinear Schrödinger equations with general Choquard type nonlinearity

In this paper, we study the following Choquard type quasilinear Schrödinger equation: −Δu+u−Δ(u2)u=(Iα∗G(u))g(u),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\Delta u+u-\Delta \bigl(u^{2}\bigr)u=\bigl(I_{\alpha }*G(u) \bigr)g(u),\quad x\in {\mathbb{R}}^{N}, $$\end{document} where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document}, 0<α<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha <N$\end{document}, and Iα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\alpha }$\end{document} is a Riesz potential. By using the minimization method developed by (Tang and Chen in Adv. Nonlinear Anal. 9:413–437, 2020; Willem in Minimax Theorems, 1996), we establish the existence of ground state solutions with general Choquard type nonlinearity. Our results extend the results obtained by (Chen et al. in Appl. Math. Lett. 102:106141, 2020).


Introduction
This article is concerned with the following quasilinear Schrödinger equation: u + uu 2 u = I α * G(u) g(u), x ∈ R N , (1.1) where N ≥ 3, 0 < α < N , I α is a Riesz potential (see [16]), and g : R N → R satisfies (g 1 ) g ∈ C(R, R); (g 2 ) there exists C > 0 such that It is well known that the existence of solitary wave solutions for the following quasilinear Schrödinger equation is a hot problem i∂ t z =z + W (x)zψ |z| 2 zl |z| 2 l |z| 2 z, (1.2) where z : R×R N → C, W : R N → R is a given potential, l : R → R and ψ : R N ×R → R are suitable functions. For various types of l and ψ, the quasilinear equation of the form (1.1) has been derived from models of several physical phenomena. For physical background, the readers can refer to [1,9,11,15] and the references therein. If we set the variable z(t, x) = exp(-iLt)u(x), where L ∈ R and u is a real function, then so many papers focused on standing wave solutions for (1.2). The readers can refer to [5,8,12,13,20] and the references therein. As for Choquard type quasilinear Schrödinger equation, there are few papers except for [3,4,21]. In [21], a class of quasilinear Choquard equations has been considered via the perturbation method developed by [13], and they showed the existence of positive solution, negative solution, and multiple solutions. Furthermore, the authors [4] established the existence of positive solutions with the periodic potential or bounded potential. In [3], the authors proved the existence of ground state solutions via Jeanjean's monotonic technique [10]. For the following Choquard equation with a local nonlinear perturbation under some suitable conditions on V , the authors proved the existence of ground state solutions without super-linear conditions near infinity or monotonicity properties on f and g in [6]. To our knowledge, there are no articles to prove the existence of ground state solutions for (1.1) with general Choquard type nonlinearity. In this paper, motivated by [3,4,6,21], we consider the existence of ground state solutions with the Berstycki-Lions conditions, which originated from [2]. To prove our results, we use the minimization method developed by Tang [18] to prove the existence of ground state solutions.
Next, the energy functional associated with (1.1) is given by To our aim, if we choose the variable u = f (v) in [7,12], then (1.1) reduces to where f : [0, +∞) → R is given by f (t) = Before stating our results, we need to define the set Q = {v ∈ H 1 (R N )\{0} : P(v) = 0}, where P is given in Lemma 2.2. Now, we give our result in the following.

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1. Next, let us recall some properties of the variables f : R → R. These properties have been proved in [7,12]. ([7, 12]) The function f (t) and its derivative satisfy the following properties: (1) f (t)/t → 1 as t → 0; (2) f (t) ≤ |t| for any t ∈ R; (3) f (t) ≤ 2 1 4 √ |t| for all t ∈ R; (4) By the standard argument in [16,19], we have the following Pohozaev type identity.
Proof By using (g 4 ) and the method in [17] and [18], it follows that Θ = ∅. Next, for any v ∈ H 1 (R N )\{0}, it follows from P(v) ≤ 0 that which shows that v ∈ Θ. The proof is completed.

Lemma 2.6
Assume that (g 1 )-(g 4 ) hold. Then, for any v ∈ Θ, there exists unique It is easy to check that lim t→0 Γ (t) = 0, Γ (t) > 0 for t > 0 enough small and Γ (t) < 0 for t large. Thus max t>0 Γ (t) is achieved at some t v > 0 such that Γ (t v ) = 0 and v t v ∈ Q.
Next, we will prove the uniqueness. For any given v ∈ Θ, if there exist t 1 , t 2 > 0 such that v t 1 , v t 2 ∈ Q. Thus P(v t 1 ) = P(v t 2 ) = 0. Therefore, we have which shows that t 1 = t 2 . Thus t v > 0 is unique for v ∈ Q. This completes the proof.

Lemma 2.7
Assume that (g 1 )-(g 3 ) hold, then Q = ∅ and Proof This result is a consequence of Corollary 2.4, Lemma 2.5, and Lemma 2.6. The proof is completed.
By a standard argument in [19], we can get the following Brezis-Lieb lemma.

Lemma 2.8 Assume that
and
(ii) For any v ∈ Q, from Lemma 2.2, we have This completes the proof.
Proof Let {v n } ⊂ Q be a minimizer for c, that is, P(v n ) = 0 and Φ(v n ) → c as n → ∞. By (2.4), one has By the Sobolev inequality, Lemma 2.1-(5), it follows that From (2.5), we infer that there exists C > 0 such that R N v 2 n ≤ C. Up to a subsequence, there exists v 0 ∈ H 1 (R N ) such that v n v 0 in H 1 (R N ), v n → v 0 in L r loc (R N ) for r ∈ [2, 2 * ) and v n → v 0 a.e. on R N . Now, we claim that there exist δ > 0 and {y n } ⊂ R N such that B 1 (y n ) |v n | 2 > δ. Assume by contradiction, by Lion's concentration compactness lemma in [19], that v n → 0 in L r (R N ) for 2 < r < 2 * . Moreover, by P(v n ) = 0, we know that as n → +∞. This is a contradiction. Thus there exist δ > 0 and {y n } ⊂ R N such that B 1 (y n ) |v n | 2 > δ. Setv n (x) = v n (x + y n ). Then v n = v n . Thus, up to a subsequence, there existsv 0 ∈ H 1 (R N )\{0} such thatv n v 0 in H 1 (R N ),v n →v 0 in L r loc (R N ) for r ∈ [2, 2 * ), andv n →v 0 a.e. on R N . By translation invariance, one has If there exists a subsequence {w n i } of {w n } such thatw n i = 0, then up to a subsequence, we have Next, we assume thatw n = 0. We claim that P(v 0 ) ≤ 0. Otherwise, if P(v 0 ) > 0, it follows from (2.7) that P(w n ) < 0 for n large. By virtue of Lemma 2.6, there exists t n > 0 such that (w n ) t n ∈ Q. By (2.7) and Lemma 2.2, we get which is a contradiction due to R N |∇v 0 | 2 > 0. Thus P(v 0 ) ≤ 0. Sincev 0 = 0, in view of Lemma 2.6, there exists t 0 > 0 such that (v 0 ) t 0 ∈ Q. By Lemma 2.3 and the weak semicontinuity of norm, we have which implies that (2.8) holds. The proof is completed.
By a standard argument in [14,18,19], we can get the following lemma. This completes the proof.