Existence of stable standing waves for the nonlinear Schrödinger equation with mixed power-type and Choquard-type nonlinearities

<abstract><p>The aim of this paper is to study the existence of stable standing waves for the following nonlinear Schrödinger type equation with mixed power-type and Choquard-type nonlinearities</p>

<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi+\Delta \psi+\lambda | \psi|^q \psi+\frac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^N}\frac{| \psi|^p}{|x-y|^\mu|y|^\alpha}dy\right)| \psi|^{p-2} \psi = 0, $\end{document} </tex-math></disp-formula></p>

<p>where $ N\geq3 $, $ 0 < \mu < N $, $ \lambda > 0 $, $ \alpha\geq0 $, $ 2\alpha+\mu\leq{N} $, $ 0 < q < \frac{4}{N} $ and $ 2-\frac{2\alpha+\mu}{N} < p < \frac{2N-2\alpha-\mu}{N-2} $. We firstly obtain the best constant of a generalized Gagliardo-Nirenberg inequality, and then we prove the existence and orbital stability of standing waves in the $ L^2 $-subcritical, $ L^2 $-critical and $ L^2 $-supercritical cases by the concentration compactness principle in a systematic way.</p></abstract>

The Eq (1.1) has several physical origins and backgrounds, which applied in various modeling scenarios arising from phenomena in science and engineering and depended on different parameter configuration, see, e.g. [20,21]. In the mathematical case λ = 0, α = 0 and p = 2, the Eq (1.1) reduces to the well-known Hartree equation, in which this type Schrödinger equations have been studied in [4,13,14] by considering the corresponding Cauchy problem. In the physical case N = 3, λ = 0, α = 0, p = 2 and µ = 2, it was introduced by Pekar in [33] to describe the quantum theory about the polaron at rest in mathematical physics. After then, Lions in [27] used it to describe an electron trapped in its own pole. In a way, it approximated to the Hartee-Fock theory about one component plasma. Afterwards, this equation was proposed by Penrose in [31,32] as a model of self-gravitating matter and usually called as the Schrödinger-Newton equation.
Recently, this type of equation has been studied extensively in [2,5,[9][10][11]19,24,29,30,35,38,39,44]. Equation (1.1) admits a class of special solutions, which are called standing waves, namely solutions of the form ψ(t, x) = e iωt u(x), where ω ∈ R is a frequency and u ∈ H 1 (R N ) is a nontrivial solution satisfying the elliptic equation (1. 2) The Eq (1.2) is variational, whose action functional is defined by where the corresponding energy functional is defined on H 1 (R N ) by To begin with, we shall focus on the existence of ground state and recall this definition. where S (c) := {u ∈ H 1 (R N ) : u 2 L 2 = c}. Subsequently, for the evolutional type equation (1.1), one of the most important problems is to study the stability of standing waves, which is defined as follows.
Otherwise, we say that the standing wave is unstable.
Generally, there are two major methods in the research of the orbital stability of standing waves. The first one is the Grillakis-Shatah-Strauss theory about general stability/instability criterion in [16]. As a matter of fact, if we assume certain spectral properties of the linearization of (1.2) about u ω , the criterion means that the standing wave e iωt u ω (x) is orbitally stable when ∂ ∂ω u ω 2 L 2 > 0 or unstable when ∂ ∂ω u ω 2 L 2 < 0. Moreover, it also turns out that this method is extremely useful in the case of homogeneous nonlinearities. In this paper, however, we consider the non-homogeneous Schrödinger equation with mixed power-type and Choquard-type nonlinearities. On the one hand, it is difficult for us to verify some properties of the spectrum. On the other hand, the sign of ∂ ∂ω u ω 2 L 2 is hard to be verified for the Eq (1.1). Therefore, this method might be hard to work, see, e.g. [25,34].
The other is the idea introduced by Cazenave and Lions in [3], which constructs orbitally stable standing waves to (1.1) is to consider the constrained minimization problems. For this method, we know that it only makes use of the conservation laws and the compactness of any minimizing sequences. Therefore, this method is quite general and may be applied to many situations. According to the idea, we naturally obtain the stability of the set of the constrained energy minimizers, and then we recall the following definition, as introduced in [3].
In view of the Definition 1.3, in order to study the stability, we require that the solution of (1.1) exists globally, at least for initial data ψ 0 sufficiently close to M. According to the results, all solutions for the nonlinear Schrödinger equation exist globally in the L 2 -subcritical case. Therefore, the stability of standing waves has been studied extensively in [2,5,8,29]. In the L 2 -supercritical case, however, according to the local well-posedness theory, the solution of NLS with small initial data exists globally, and the solution may blow up in finite time for some large initial data. Therefore, the existence of stable standing waves in this case is of particular interest. Meanwhile, this type of problems have been considered in [18,19,35] by studying the corresponding minimization problem recently.
At this point, the nonlinear Schrödinger equation have attracted much attention. When α = 0, Li and Zhao [29] showed the existence and orbital stability of standing waves in the mass subcritical case and mass critical case. Chen and Tang [2] obtained the existence of normalized ground states. The ground states of the NLS equation with combined power-type nonlinearities was studied by Jeanjean in [19] and Soave in [35,36]. The related content with Choquard-type nonlinearities was obtained by Feng and Chen in [9,12]. In the case N = 3, λ = 0, α = 0, p = 2 and µ = 2, the existence and orbital stability of standing waves were proved by Cazenave and Lions in [3].
From a mathematical point of view, however, the Choquard-type equation (1.2) also stimulated a lot of interest see, e.g. [6,7,17,[41][42][43]. In the case λ = 0, Du, Gao and Yang [5] studied the existence of positive ground state in the energy subcritical and the energy critical cases, established the regularity and symmetry by the moving plane method in integral forms. Furthermore, the existence and uniqueness of positive solutions was proved by Lieb [22] and Lions [27], and the orbital stability of generalized Choquard-type equation was obtained by Wang, Sun and Lv [40].
In this paper, the study of the existence and stability of standing waves for (1.1) with α > 0 in the energy space H 1 is of particular interest, in which the time of existence only depends on the H 1 -norm of initial data. Therefore, by the Gagliardo-Nirenberg inequality and the concentration compactness principle in the study of orbital stability of standing waves, see, e.g. [3,14,15,17,26,44], we can obtain the following main results: In the mass subcritical case, i.e., 2 − 2α+µ N < p < 2+2N−2α−µ N and c > 0, or in the mass critical case, i.e., p = 2+2N−2α−µ N and 0 < c < Q p 2 L 2 , where Q p is a ground state to the elliptic equation it is easy for us to see that the energy functional is bounded from below on S (c). In particular, for α = 0, in view of (1.4), the Riesz potential I µ : R N → R is defined by Therefore, applying the concepts by Cazenave and Lions in [3], we consider the following constrained minimization problem m(c) := inf However, compared with the work for the classical Schrödinger equation, there are two major difficulties in the analysis of stable standing waves. One is that the Eq (1.2) does not enjoy the scaling invariance and the space translation invariance due to the inhomogeneous nonlinearity |x−y| µ |y| α dy |u| p−2 u, the other is that the nonlinear term with a convolution is difficult to handle. Therefore, the usual methods cannot work. In order to overcome these difficulties, we need to prove the boundedness of the translation sequence {y n }, and then apply it to prove the compactness of all minimizing sequences for (1.5). Based on the result, we can obtain the existence of minimizers for the minimization problem (1.5) and the stability of standing waves. Theorem 1.4. Let N ≥ 3, 0 < µ < N, λ > 0, α ≥ 0, 2α + µ ≤ N and 0 < q < 4 N . Assume one of the following conditions hold: In the mass supercritical case, i.e., 2+2N−2α (1.6) In view of (1.6), we can obtain that E(u s ) → −∞ as s → ∞. Therefore, we cannot study the existence of stable standing waves for (1.1) by considering the global minimization problem (1.5). Applying the concepts by Jeanjean in [19], Luo and Yang in [28], we consider the following constrained local minimization problem m(c) := inf where V(c) := S (c) ∩ B r 0 = {u ∈ S (c) : ∇u 2 L 2 < r 0 } for r 0 > 0 with c ∈ (0, c 0 ), and B r 0 is defined by More precisely, we can obtain the property that where ∂V(c) := {u ∈ S (c) : ∇u 2 L 2 = r 0 }. However, the energy functional of (1.3) does not keep invariant by translation due to the inhomogeneous nonlinearity 1 Similarly, in order to prove the compactness of all minimizing sequences for the minimization problem (1.7), we can solve it by proving the boundedness of any translation sequences. As consequence, we can obtain the existence of minimizers for the minimization problem (1.7) and the stability of standing waves.
Then there exists a c 0 > 0 with c ∈ (0, c 0 ) such that the following conclusions hold: (2) The set M c is orbitally stable.
This paper is organized as follows. In section 2, we give some preliminaries. Next, we obtain the best constant of the Gagliardo-Nirenberg inequality (2.5). In section 3, we prove the Theorem 1.4. In section 4, we give some properties for (1.1) in the mass supercritical case. In section 5, we prove the Theorem 1.5. In section 6, we make a summary for this paper.
Notation: Throughout this paper, we use the following notation. C > 0 stands for a constant that may be different from line to line when it does not cause any confusion.
s . B R (y) denotes the ball in R N centered at y with radius R.

Preliminaries
In this section, we we will collect some preliminaries, and then we obtain the best constant of the Gagliardo-Nirenberg inequality (2.6).
Moreover, there are conservations of mass and energy, Next, we can establish the following Gagliardo-Nirenberg inequality related to (1.2) and the concentration compactness principle.
N−2 , then the following sharp Gagliardo-Nirenberg inequality where Q q is a ground state solution of the elliptic equation −∆Q q + Q q = |Q q | q Q q . In particular, in the L 2 -critical case, i.e., q = 4 where λ > 0 is fixed. Then there exists a subsequence {u n k } ∞ k=1 satisfying one of the three possibilities: (iii) (Dichotomy) There exists σ ∈ (0, λ) and u (1) n k , u (2) n k bounded in H 1 (R N ) such that: Then, there exists a constant C(α, µ, N, p, r), independent of u, v, such that where C > 0 is a constant depending only on N, α, µ and s.
(2) By the Lemma 2.4 and the Sobolev embedding theorem, we can obtain that where C > 0 is a constant depending only on N, α, µ and p.
By applying the idea of M.Weinstein [37], the best constant for the generalized Gagliardo-Nirenberg inequality (2.6) can be obtained by considering the existence of the following functional More precisely, we can obtain the following theorem.
The best constant in the generalized Gagliardo-Nirenberg inequality is defined by where Q p is a ground state solution of the elliptic equation (1.4).
In particular, in the L 2 -critical case, i.e., p To start with, by Lemma 2.4 and applying the interpolation inequality and Sobolev imbedding, we can obtain that Based on the above results, the functional (2.4) is well-defined. Thus, we consider a minimizing sequence {u n } and the following variational problem By the Gagliardo-Nirenberg inequality, we have J > 0. Similarly, we set a minimizing sequence {v n } ∞ n=1 , which is defined by v n (x) = µ n u n (λ n x) with By the Schwarz symmetrization properties, namely we may assume that v n is spherically symmetric and satisfies v * n H 1 ≤ v n H 1 . Consequently, there exists a subsequence, which we still denote by On the basis of the standard variational principle, if w ∈ H 1 (R N ), we have Then, we can obtain that v satisfies the following elliptic equation Then, we can establish the following Pohozaev identity (see Lemma 3.1 in [5]) related to (1.4). Multiplying (1.4) by Q p and by x · ∇Q p , and integrating by parts, we have and From these identities, we can get the following relations Having said all of above, we derive the best constant The conclusion was arrived.

Proof of Theorem 1.4
In this section, we prove the Theorem 1.4 in seven steps.
Step 1. We prove that the minimization problem (1.5) is well-defined and every minimization sequence of (1.5) is bounded in H 1 (R N ). By the definition of E(u) and applying the Lemma 2.2 and the Young inequality, see, e.g. [29,39], we have for any ε > 0 and u ∈ S (c).
In the case 2 − 2α+µ In the case By (3.1) and u L 2 < Q p L 2 , it follows from the Theorem 2.6 that Therefore, E(u) has a lower bound and the variational problem (1.5) is well-defined. Moreover, it is easy for us to see that every minimization sequence of (1.5) is bounded in H 1 (R N ).
Step 2. We do the scaling transform of the energy functional (1.3) for s > 0 sufficiently small. Based on the above analysis, in the case 2 − 2α+µ , in view of (1.6), we can find an s > 0 sufficiently small such that E(u s ) < 0.
Next, we choose {u n } ∞ n=1 ⊂ S (c) be a minimizing sequence bounded in H 1 (R N ) satisfying Then, there exists a subsequence {u n k } ∞ k=1 such that one of the three possibilities in Lemma 2.3 holds.
Step 3. We prove that the vanishing case in Lemma 2.3 does not occur. If not, by Lion's lemma, we have u n k → 0 in L s (R N ) as k → ∞ for all s ∈ (2, 2N N−2 ). Hence, which contradicts m(c) < 0. Hence, the vanishing does not occur.
Hence, we can obtain that where . Afterwards, we suppose by contradiction that (iii) in Lemma 2.3 holds. Thus, there exist {u (1) n k } and {u (2) n k } such that d n k = dist{Supp u (1) n k , Supp u (2) n k } → ∞, and as k → ∞. Similar to the proof of the Brézis-Lieb Lemma [1], we know that 2α+µ (R N ) as n → ∞. Hence, by some basic calculation we can obtain that By the Lemma 2.3, we know that Supp u (1) We consequently obtain that If we denoteũ n k (·) = u n k (· + y n k ), then there existsũ satisfying R N |ũ(x)| 2 dx = λ, namelỹ u n k ũ in H 1 (R N ) andũ n k →ũ in L s (R N ) for all s ∈ [2, 2N N − 2 ).
Step 5. We prove that the compactness case in Lemma 2.3 will occur. We firstly prove that the sequence {y n k } ∞ k=1 is bounded. Indeed, if it was not true, then up to a subsequence, we assume that |y n k | → ∞ as n → ∞. We consequently deduce that as k → ∞, which yields m(c) ≥ m ∞ (c). In fact, by the definition of m ∞ (c), we know that m ∞ (c) is attained by a nontrivial function v c , which yields We can see thatũ is a minimizer of m ∞ (c), and then we can obtain Accordingly, {y n k } ∞ k=1 is bounded, and up to subsequence, we assume that lim k→∞ y n k = y 0 . We consequently deduce that Therefore, E(u) = m(c) and u n k → u in H 1 (R n ) as k → ∞. This completes the proof.
Step 6. We prove that the Cauchy problem (1.1) admits a global solution ψ(t) with Indeed, by Lemma 2.1, we know that it suffices to bound ∇ψ(t) L 2 in the existence time. By Lemma 2.2, Theorem 2.6, the conversation law and the Young inequality, we have Similar to the step 1, in the case 2 − 2α+µ In the case p = 2+2N−2α−µ N , we have The above argument implies the boundedness of ∇ψ(t) 2 L 2 since ψ(t) L 2 = ψ(0) L 2 < Q p L 2 . Then we come to the conclusion.
Step 7. We prove that the set M c is orbitally stable. We firstly observed that the solution ψ of (1.1) exists globally, then argue by contradiction that there exist constant ε 0 > 0 and a sequence and there exists {t n } ∞ n=1 ⊂ R + such that the corresponding solution sequence ψ n (t n ) of (1.1) satisfies sup By the conservation of mass and energy, we have Similarly, by the argument above, we can see that {ψ n (t n )} ∞ n=1 is bounded in H 1 (R N ). Hence, ψ n (t n ) L 2 and ψ n 2 L 2 = c. From the above results, we have lim Hence, {ψ n (t n )} ∞ n=1 is a minimizing sequence of (1.5). By the analysis above, there existsṽ ∈ M c satisfiesψ n →ṽ in H 1 (R N ). (3.11) By the definition ofψ n , we knowψ We consequently obtain that ψ n (t n ) →ṽ in H 1 (R N ), and which contradicts (3.8). This completes the proof.

The supercritical case
By the definition of E(u) and applying the Lemma 2.2 and Theorem 2.6, we have where C 1 = C(q), C 2 = C α,µ,p . First of all, we define the following function of two variables, namely Now, according to the configurations of parameters above, we note that And then, substitute the notation into the function, we have In the L 2 -supercritical case, however, we notice that if N ≥ 3, 0 < µ < N, Lemma 4.1. The function g c (r) has a unique global maximum and the maximum value satisfies Proof. By the definition of g c (r), we can obtain by some calculation that g c (r) = − β 1 2 Hence, there has unique solution of the equation g c (r) = 0, namely Moreover, considering in the analysis of g c (r) we know that g c (r) → −∞ as r → 0 and g c (r) → −∞ as r → +∞. Therefore, we can deduce that r c is the unique global maximum point of g c (r), namely In view of (4.2), we can obtain that max r>0 g c 0 (r c 0 ) = 0, and hence the lemma follows.
. Then for any c 2 ∈ (0, c 1 ], we have that Proof. It is shown that c → f (·, r) is a non-increasing function, and then we have By some basic calculations, β 1 + β 2 = q > 0, and taking into account we have Moreover, we observe that if g c 2 (r ) ≥ 0 and g c 2 (r ) ≥ 0 then f (c 2 , r) = g c 2 (r) ≥ 0 for any r ∈ [r , r ]. (4.7) Indeed, there exists a local minimum point on (r , r ) when g c 2 (r) < 0 for r ∈ [r , r ], and which contradicts to the fact that g c 2 (r) has unique critical point with global maximum (see Lemma 4.1). By (4.5) and (4.6), we can choose r = c 2 c 1 r 1 and r = r 1 , and hence the lemma follows. By the Lemmas 4.1 and 4.2, we can obtain that f (c 0 , r 0 ) = 0 and f (c, r 0 ) ≥ 0 for all c ∈ (0, c 0 ) and r 0 := r c 0 > 0. According to the above results, we have the following lemma. Proof. Firstly, we know that the sequence {c n } ⊂ (0, c 0 ) satisfies c n → c. By the definition of m(c n ), we can obtain that there exists u n ∈ V(c n ) such that E(u n ) < 0 and E(u n ) ≤ m(c n ) + ε for all ε > 0 small enough. As mentioned above, by the definition of z n , we can obtain that and then, we can write it as At this point, by the definition of V(c), we can obtain that ∇u n 2 L 2 < r 0 for u ∈ V(c). Moreover, we also know that u n q+2 L q+2 and R N R N |u| p |u| p |x| α |x−y| µ |y| α dxdy are uniformly bounded, then m(c) ≤ E(u n ) + o n (1) as n → ∞. (4.9) In view of (4.8) and (4.9), we have m(c) ≤ m(c n ) + ε + o n (1), then there exists u ∈ V(c) such that E(u) < 0 and E(u) ≤ m(c) + ε for all ε > 0 small enough.
Similar to the argument above, we denote u n := c n c u ∈ V(c n ) ⊆ S (c n ), by the fact that c n → c and E(u n ) → E(u) for u n ∈ V(c n ), then (4.10) Therefore, we conclude that m(c n ) → m(c) for all ε > 0 small enough, and hence the lemma follows.
Lemma 4.4. Let {v n } ∞ n=1 ⊂ B r 0 be such that v n L q+2 → 0. Then, there exist a constant γ 0 > 0 such that E(v n ) ≥ γ 0 ∇v n 2 L 2 + o n (1). Proof. As a matter of fact, by the Theorem 2.6 we obtain that Hence, by the fact that f (c 0 , r 0 ) = 0, we have that

Proof of Theorem 1.5
In this section, we prove the Theorem 1.5 in seven steps.
Step 1. We prove that the minimization problem (1.7) is well-defined. First of all, we have ∇u 2 L 2 = r 0 for all u ∈ ∂V(c). Then, in view of (4.1), we can get Similarly, in view of (1.6), we can get φ u (s) := E(u s ) < 0 for all s > 0 small enough.
As mentioned above, we obtain that Therefore, E(u) has a lower bound and the variational problem (1.7) is well-defined.
Step 2. We prove that the ground state is local minimizer of E(u) contained in V(c) when m(c) is reached. Firstly, we assume that u is a critical point of E(u), its restriction u ∈ S (c) belong to the set Moreover, by some basic calculations we have the derivative of φ v , namely Similarly, we observe the fact that if ∇v L 2 = 1 with v ∈ S (c) so that u = v s with u ∈ S (c) for s ∈ (0, ∞). As a matter of fact, the ground states is contained in the set Q c . In view of (5.2), if w ∈ Q c and v ∈ S (c) satisfies ∇v L 2 = 1, so that w = v s 0 , E(w) = E(v s 0 ) and d ds E(v s )(s 0 ) = 0 for s 0 ∈ (0, ∞). Just by the properties of derivatives, s 0 ∈ (0, ∞) is a zero of the function φ v (s).
By the definition of ∂V(c), however, when v s ∈ ∂V(c) we can easily acquire that Hence, s 1 > 0 is the first zero of d ds E(v s ), and it is the local minima satisfying E(v s 1 ) < 0 for v s 1 ∈ V(c). On the other hand, when v s ∈ ∂V(c) we also have E(v s 1 ) < 0, E(v s ) ≥ 0 and Hence, s 2 > s 1 is the second zero of E(v s ), and it is the local maxima satisfying E(v s 2 ) ≥ 0 and m(c) ≤ E(v s 1 ) < E(v s 2 ). In particular, v s 2 cannot be a ground state if m(c) is reached.
To sum up, φ v has at most two zeros, that is equal to the function s → φ u (s) s has at most two zeros, which yields that s 0 = s 1 and ω = v s 0 = v s 1 ∈ V(c). According to the basic calculations, we obtain that From what has been discussed above, we know that β 1 < 0 and β 3 > 0, then h (s) = 0 has a unique solution, and h(s) = 0 has indeed at most two zeros. Moreover, the solutions were local minimizer contained in V(c).
Step 3. We prove that the vanishing case does not occur. If not, we assume that First of all, let {u n } ∞ n=1 ⊂ B r 0 is bounded in H 1 (R N ) be such that u n 2 L 2 → c and E(u n ) → m(c) for all c ∈ (0, c 0 ). By Lions' lemma, we deduce that u n L q+2 → 0 as n → ∞. At this point, by the Lemma 4.4, we have that E(u n ) ≥ o n (1), which is a contradiction with m(c) < 0.
Step 5. We prove that the compactness case will occur. By a similar argument above, using the Lemma 2.3 and Step 5 of the proof of Theorem 1.4, we know that the sequence {y n } ⊂ R N is bounded, and up to a sequence, we assume that y n → y 0 as n → ∞. We consequently deduce that First of all, we denote w n (x) := u n (x − y n ) − u c (x), we need to prove that the compactness holds, i.e., Again, by the definition of u n and the analysis of w n , we can obtain that As mentioned, we can obtain that For this reason, in view of (5.7) and (5.8), we notice that any term in E fulfills the splitting properties of Brézis-Lieb [1]. Consequently, By using the fact that {y n } is bounded and the translation invariance holds, we have On the one hand, in order to prove the compactness holds, we firstly prove that w n 2 L 2 → 0. In view of (5.7), if we note c 1 := u c 2 L 2 > 0 so that the conclusion arrived when c 1 = c. Instead, if we argue by contradiction with c 1 < c, by the analysis of (5.7) and (5.8), we have While in the mass supercritical case, by the definition of w n , we have Recalling E(u n ) → m(c) and in view of (5.9), then In context, by Lemma 4.3 we know that the map c ∈ (0, c 0 ) → m(c) is continuous. Thus, in view of (5.7), we can deduce that u c ∈ V(c 1 ) and It is impossible to m(c) > m(c). Consequently, we conclude that u c 2 L 2 = c and w n 2 L 2 → 0. On the other hand, we next prove that ∇w n 2 L 2 → 0. With all that said, in view of (5.8), we can deduce that {w n } ∞ n=1 ⊂ B r 0 is bounded in H 1 (R N ). By the Gagliardo-Nirenberg inequality of Lemma 2.2 and Theorem 2.6, we can obtain that w n q+2 L q+2 → 0 and R N R N |w n | p |w n | p |x| α |x−y| µ |y| α dxdy → 0. Consequently, by the Lemma 4.4, we have E(w n ) ≥ γ 0 ∇w n 2 L 2 + o n (1) where γ 0 > 0. (5.11) At the end of the part, due to u n u c in H 1 (R N ) with u c ∈ V(c), in view of (5.9), we consequently deduce that E(u c ) ≥ m(c) and E(w n ) ≤ o n (1), namely ∇w n 2 L 2 → 0. Above all, w n → 0 in H 1 (R N ) and we come to the conclusion.
Step 6. We prove that the Cauchy problem (1.1) admits a global solution ψ(t) with ψ(0, and ψ 0 ∈ H 1 (R N ). Firstly, we denote the right hand of (5.1) by A. Since the energy E(u) is the continuous function with respect to u ∈ H 1 (R N ), we deduce from E(u) = m(c) < A that there is a δ > 0 such that ψ 0 − u H 1 < δ for ψ 0 ∈ H 1 (R N ), and we have E(ψ 0 ) < A.
Step 7. We prove that the set M c is orbitally stable. We argue by contradiction, i.e., we assume that there is ε 0 > 0, a sequence of initial data {ψ 0,n } ⊂ H 1 (R N ) and a sequence {t n } ⊂ R satisfy the maximal solution ψ n (t) with ψ n (0) = ψ 0,n such that which implies that {ψ n } is a minimizing sequence for (1.7). Thanks to the compactness of all minimizing sequence of (1.7), there is aũ ∈ M c satisfiesψ n →ũ in H 1 (R N ). Moreover, by the definition ofψ n , it follows thatψ n → ψ n (t n ) in H 1 (R N ). Consequently, we have ψ n (t n ) →ũ in H 1 (R N ), which contradicts to (5.12). This completes the proof.

Conclusions
In this work, we study the stability of set of energy minimizers in the mass subcritical, mass critical and mass supercritical cases. Due to appearance of the inhomogeneous nonlinearity 1 |x| α R N |u| p |x−y| µ |y| α dy |u| p−2 u, the non-vanishing of any minimizing sequence is hard to exclude. By a rather delicate analysis, we can overcome this difficulty by proving the boundedness of any translation sequence. To the best of our knowledge, there are no any results about instability or strong instability. However, for its mathematical interest, these problems will be the object of a future investigation.