Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

We study the coupled Choquard type system with lower critical exponents {−Δu+λ1(x)u=μ1(Iα∗|u|N+αN)|u|αN−1u+β(Iα∗|v|N+αN)|u|αN−1u,x∈RN,−Δv+λ2(x)v=μ2(Iα∗|v|N+αN)|v|αN−1v+β(Iα∗|u|N+αN)|v|αN−1v,x∈RN,u,v∈H1(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -\Delta u+\lambda _{1}(x)u=\mu _{1}(I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u+\beta (I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u,\quad x\in {\mathbb{R}}^{N}, \\ -\Delta v+\lambda _{2}(x)v=\mu _{2}(I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v+\beta (I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v,\quad x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{cases} $$\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}$N\ge 3$\end{document}, μ1,μ2,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu _{1}, \mu _{2}, \beta >0$\end{document}, and λ1(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{1}(x)$\end{document}, λ2(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{2}(x)$\end{document} are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{1}$\end{document}, λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda _{2}$\end{document}.

Mathematically, Choquard type equations have received considerable attention in the past few years, see [1, 3-5, 7, 8, 11, 15-17] and the reference therein for scale equations. There are also some results concerned with solutions of a nonlinearly coupled Choquard system. In [21], Wang and Shi proved the existence of positive solutions of ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u + λ 1 u = μ 1 (I α * |u| 2 )u + β(I α * |v| 2 )u, x ∈ R N , (1.2) for λ 1 , λ 2 > 0 and β ∈ (-∞, χ 0 ) ∪ (min{λ 2 μ, λ 1 2 ν}, +∞), where λ = λ 2 /λ 1 and χ 0 > 0 depends on μ 1 , μ 2 , λ. Particularly, when λ 1 = λ 2 > 0, they showed that system (1.2) has a positive ground state ( √ k 0 w 0 , √ l 0 w 0 ), where (k 0 , l 0 ) is the solution of ⎧ ⎨ ⎩ μ 1 k + βl = 1, and w 0 is a positive ground state of In [22], Wang and Yang established the existence and nonexistence of normalized solutions of system (1.2) with trapping potentials. In [20], Wang obtained the multiplicity of nontrivial solutions of a nonlinearly coupled Choquard system with general subcritical exponents and perturbations. For a Choquard system with upper critical exponents, You, Wang, and Zhao [25,26] derived the existence of a positive ground state of the following system: (1.5) where N ≥ 5, is a bounded smooth domain in R N , -λ 1 ( ) < λ 1 , λ 2 < 0, and λ 1 ( ) represents the first eigenvalue of -on with the Dirichlet boundary condition. More precisely, they obtained that system (1.5) has a positive ground state if For the special case -λ 1 ( ) < λ 1 = λ 2 < 0, they proved that system (1.5) has a positive ground state ( k w * , l w * ) if where w * is a positive ground state of andk,l is a solution of In the current paper, we study the nonlinearly coupled system (1.1) with lower critical exponents. Since system (1.1) with positive constant potentials has no nontrivial solution in H := H 1 (R N ) × H 1 (R N ) by the Pohozaev identity, we assume that λ 1 , λ 2 are functions dependent on x ∈ R N . We aim to prove the existence of positive ground states of system (1.1). Furthermore, for the case λ 1 (x) = λ 2 (x) := λ(x), we will introduce an approach which is different with [21,25,26] to prove that system (1.1) has a positive ground state of the form (kw, lw), where w is a positive ground state of For this purpose, we assume that Note that under assumptions (C1) and (C2), the scale equation has a ground state w i , i = 1, 2 (see [16,Theorem 3,Theorem 6]). Moreover, we may assume that w i is positive since |w i | is also a ground state of (1.9). Clearly, system (1.1) has a trivial solution (0, 0) and two semi-trivial solutions (w 1 , 0) and (0, w 2 ) for all β ∈ R. Here we deal with the nontrivial solution, that is, a solution (u, v) of (1.1) with u ≡ 0 and v ≡ 0. Denote R N ·dx by · for simplicity, and define the functional I : H → R corresponding to system (1.1) by 3] to our case, we can prove that system (1.1) has a ground state of the form (kw, lw) only if β ≥ α N max{μ 1 , μ 2 }. In the current paper, we use an alternative approach inspired by [24], which is based on studying the minimum point of g(s), and we show that system (1.1) possesses a ground state of this form for all β > 0. Remark 1.3 The method we adopted in the proof of Theorem 1.1 is also valid for the upper critical system (1.5). As we mentioned previously, system (1.5) has a ground state of the form (kw * , lw * ) if N ≥ 5, -λ 1 ( ) < λ 1 = λ 2 < 0, and β ∈ (max{μ 1 , μ 2 }, +∞) for α = N -4 (see Theorem A.1 in Appendix). Although our approach can only deal with the case β > max{μ 1 , μ 2 } for α = N -4, in the case α ∈ (0, N -4), the existence of a ground state of (kw * , lw * ) type is obtained for all β > 0.
In the proof of Theorem 1.4, we need to give an accurate estimate of the least energy so as to overcome the lack of compactness and show that both components of the solution we obtained are nontrivial. For this purpose, some results of equation (1.9) will be used. Denote the functional associated with (1.9) by and set Then, from [16, Theorem 3,Theorem 6] and some calculation, we see that B i is attained and (1.12) By [9, Theorem 3.1], S 1 has a unique minimizer (1.13) We should also study the minimizing problem where L = L 2 (R N ) × L 2 (R N ). Problem (1.14) can be seen as an extension of the classical problem (1.12). By a similar approach as in the proof of Theorem 1.1, we obtain the following result.

Theorem 1.5
If β > 0, then S 0 = g(s m )S 1 , and (s m U * , U * ) is a solution of (1.14), where g(s) is defined in (1.10) and s m is a minimum point of g(s). If β < 0, then and S 0 is not attained.
Theorem 1.5 not only plays an important role in the proof of Theorem 1.4, but also extends the classical results of [9, Theorem 3.1].

Proof of Theorem 1.1
In order to prove Theorem 1.1, we study the minimizing problem Up to multiplication by a scalar, we know that a minimizer of A is a ground state of system (1.1) for λ 1 Letting w be a solution of (1.8), we know that A 1 is attained by w. By studying a function g : R + → R defined by we are able to obtain the relationship between A and A 1 and show that A is attained.
Proof By simple calculation, we have Let (u n , v n ) ∈ H be a minimizing sequence of A, and set ξ n = τ n u n , where Then we have From (2.4) and the property of the Riesz potential that I α = I α 2 * I α 2 , we obtain (2.5)

By (2.4) and (2.5), we have
On the other hand, ∀(u, v) ∈ M, we have , that is, (t m s m w, t m w) is a positive ground state of system (1.1).

Proof of Theorem 1.5
In this section, we prove Theorem 1.5, which is essential in the proof of Theorem 1.4. Recalling the definition of U * , we have the following lemma. Proof By a similar approach as that in Lemma 2.2, we see that S 0 = g(s m )S 1 . Then the conclusion follows from Then, for |y| sufficiently large, By letting y → +∞, we get On the other hand, since β < 0, we know that Therefore, which contradicts (3.1). Thus, the conclusion holds.
Proof of Theorem 1.5 By Lemmas 3.1 and 3.2, we see that Theorem 1.5 holds.

Proof of Theorem 1.4
In this section, we define By simple calculation and analysis, we see that for any (u, v) = (0, 0), there exists t 0 > 0 such that t 0 (u, v) ∈ N and I(t 0 u, t 0 v) = max t≥0 I(tu, tv). Then, as in the proof of [23, Theorem 4.2], we know that Moreover, since M ⊂ N , we have B ≤ B. We will show that B is attained by some positive solution (u, v) of system (1.1). To begin with, we give an estimate of the upper bound of B, which is important in recovering the compactness of the Palais-Smale sequence. Recall (s m U * , U * ) defined in Theorem 1.5, and let t > 0 be the constant such that t(s m U * , U * ) ∈ N . Then, by Theorem 1.5 and direct calculation, we see that To get (4.1), it suffices to show for some b ∈ R N . By the fact that After a transformation x = b + ay, we have Then from (C2) we see that (4.2) holds for b = 0, and (4.1) follows.
Next, we show B < B i , i = 1, 2. Let w i be a positive solution of (1.9) for i = 1, 2 and t(τ ) > 0 such that By simple calculation, we get It follows that Therefore, Similarly, we have B < B 2 .
Next, we prove a Brezis-Lieb type lemma.
Proof From the Brezis-Lieb lemma [23], we know that as n → ∞. Then, according to the Hardy-Littlewood-Sobolev inequality, we have as n → ∞. Observing that and |u n -u| we see that the conclusion holds.
Proof of Theorem 1.4 According to the mountain pass theorem [23], we obtain that there is {(u n , v n )} ⊂ N satisfying It follows that for n large enough, which combined with assumption (C1) implies that {(u n , v n )} is bounded in H. Then we may assume that Using the Hardy-Littlewood-Sobolev inequality, we obtain Observing that we have, for any φ ∈ C ∞ 0 (R N ), Combining (4.5) with Lemma 4.2, we have, for n large enough, and Set C n = |∇z n | 2 + z 2 n , D n = |∇ω n | 2 + ω 2 n .
Then, by (4.8) and (1.11), we obtain which contradicts Lemma 4.1. Similarly, F n → 0 also leads to a contradiction. Thus, E n ≥ δ and F n ≥ δ for some δ > 0 and n large enough. Then there exists t n > 0 such that J (t n z n , t n ω n ), (t n z n , t n ω n ) = 0 and J(t n z n , t n ω n ) = max where the last inequality follows by Theorem 1.5. Moreover, by (4.6), we have t n → 1. Then we have which also contradicts Lemma 4.1.
(iii) v ≡ 0, u ≡ 0. By similar arguments as in case (ii), we see that B ≥ B 1 , which also contradicts Lemma 4.1.

N-2 N+α
, and w * is a positive ground state of (1.6).

Proof of Theorem
On the other hand, for any (u, v) ∈ M * , by Lemma A.2 again, we have Thus, ζ m (s * m w * , w * ) is a positive ground state of (1.5).