Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $\lambda>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $\lambda\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-\Delta u+u=u^{q-1}$ and as $\lambda\to +\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-\Delta u=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$.

as λ → 0 and λ → +∞, where N = 3 or N = 4, 2 < q ≤ p ≤ 2 * , 2 * = 2N N −2 is the Sobolev critical exponent, a > 0, b ≥ 0 are constants and λ > 0 is a parameter.In particular, we prove that in the case 2 < q < p = 2 * , as λ → 0, after a suitable rescaling the ground state solutions of (P λ ) converge to the unique positive solution of the equation −∆u + u = u q−1 and as λ → +∞, after another rescaling the ground state solutions of (P λ ) converge to a particular solution of the critical Emden-Fowler equation −∆u = u 2 * −1 .We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension N = 3 and N = 4.We also discuss a connection of our results with a mass constrained problem associated to (P λ ) with normalization constraint R N |u| 2 = c 2 .As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem.In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c > 0 is sufficiently large or sufficiently small.Our results also show that in the space dimension N = 3, there is a striking difference between the cases b = 0 and b = 0.More precisely, if b = 0, then both p0 := 10/3 and p b := 14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.

Introduction and notations
We consider the following Kirchhoff equation where N ≥ 3, 2 < q ≤ p ≤ 2 * = 2N N −2 , a > 0, b ≥ 0 and λ > 0 are parameters.For a fixed λ > 0, the corresponding to (P λ ) action functional is given by and critical points of I λ in H 1 (R N ) correspond to solutions of (P λ ).By a ground state solution of (P λ ) we understand a solution u λ ∈ H 1 (R N ) such that I λ (u λ ) ≤ I λ (u) for every nontrivial solution u of (P λ ).
In this paper we are interested in the limit asymptotic profile of the ground sates u λ of the problem (P λ ), and in the precise asymptotic behavior of different norms of u λ , as λ → 0 and λ → ∞.Of particular importance is the L 2 -mass of the groundstates which plays a key role in the analysis of stability of the standing wave solution of the timedependent NLS , cf.Lewin and Nodari [14, Section 3.2] for a discussion in the context of the local combined power NLS.
For the local prototype of (P λ ) with b = 0 the asymptotic profiles of the ground sates were studied in [1,2], where the authors considered the following nonlinear Schrödinger equation with focusing combined powers nonlinearity: where N ≥ 3 and 2 < q < p ≤ 2 * .When p = 2 * and q ∈ (2 + 4 N , 2 * ), Akahori et al. in [1] proved that for small λ > 0 the positive ground state of (1.3) is unique and non-degenerate, and as λ → 0 the unique positive ground state u λ converges after an explicit rescaling to the unique positive solution of the limit equation −∆u + u = u q−1 in R N .In [2], after a suitable implicit rescaling the authors establish a uniform decay estimate for the positive ground states u λ , and then prove the uniqueness and nondegeneracy of ground states u λ for N ≥ 5 and large λ > 0, and show that for N ≥ 3, as λ → ∞, u λ converges to a particular solution of the critical Emden-Fowler equation.Recently, for p = 2 * , Coles and Gustafson [5] proved that the radial ground state u λ is also unique and non-degenerate for all large λ > 0 when N = 3 and q ∈ (4, 2 * ).In [19], the authors studied a related problem and its connection with a mass constrained problem by using a rescaling argument and the concentration-compactness principle.See also [20] for a nonlinear Choquard type equation.
The techniques in this work (as well as in [19,20]) is inspired by [21], where the second author and C. Muratov studied the asymptotic properties of ground states for a combined powers Schrödinger equations with a focusing exponent p > 2 and a defocusing exponent q > p, and obtained a sharp asymptotic characterisation of the limit profiles of positive ground states u λ of (1.4) as λ → 0. Later, in [14], M. Lewin and S. Rota Nodari proved a general result about the uniqueness and non-degeneracy of positive radial solutions of (1.4).The nondegeneracy of the unique positive solution allowed them to refine the asymptotic results in [21] and, amongst other things, to establish the exact asymptotic behavior of M (λ) = u λ 2 2 .In particular, this implied the uniqueness of normalised energy minimizers at fixed masses in certain regimes.See also [17], where Zeng Liu and the second author extended the results in [21] to a class of Choquard type equation.
In the present paper, we study the limit asymptotic profiles of the ground sates u λ of the Kirchhoff problem (P λ ) by using a rescaling argument and the concentration-compactness principle, and obtain an explicit asymptotic expression of different norms of ground states for fixed frequency problem (P λ ).To do so, we adapt the technique developed in [19].However, additional difficulties arise since (P λ ) contains five terms and as a consequence, the Pohožaev-Nehari algebraic relations can not be resolved, in general.Fortunately, we succeed to overcome this difficulty in the case p = 2 * , but the method does not work any more for p < 2 * .In the latter case, we shall use a suitable scaling to reduce (P λ ) to the local equation (1.3) (cf.(5.2)).The disadvantage of such a rescaling is that for b = 0, the rescaled family of ground states for (P λ ) should not necessarily be a ground state for (1.3).Besides, to obtain a precise estimate of the least energy, the scaling should transform a ground state for (P λ ) into a ground state of the local equation (1.3).Generally speaking, this is not the case, which prevents us from deriving a precise energy estimate of the ground state p < 2 * , see Section 5 for a discussion.
Alternatively to the study of fixed frequency solutions of (P λ ), one can search for solutions to (P λ ) with a prescribed mass, that is for a fixed c > 0 to search for u ∈ H 1 (R N ) and λ ∈ R that satisfy where µ > 0 is a new parameter and the frequency λ ∈ R becomes a part of the unknown.The solutions of (1.5) are usually denoted by a pair (u, λ) and referred to as normalized solutions.
Normalised solutions can be obtained by searching critical points of the energy functional subject to the constraint where λ ∈ R appears a posteriori as a Lagrange multipliers.
In the local case b = 0, by rescaling we also assume a = 1.Then equation (P λ ) reduces to the classical non linear Schrödinger equation (1.7) Normalized solutions of (1.7) were studied by T. Cazenave and P.-L.Lions [3], N. Soave [24,25], L. Jeanjean et al. [10,12], L. Jeanjean and T. Le [11].The additional parameter µ > 0 is often introduced to control the unknown Lagrange multipliers λ ∈ R. Some of the results on normalized solutions to (1.7) are summarized in [15].The asymptotic behavior of normalized solution as µ varies in its range is studied in [11,24,25,26,27].We mention that in the case N ≥ 4, 2 < q < 2+ 4 N and p = 2 * , L. Jeanjean and T. Le [11] obtained a normalized solution u c of mountain pass type for small c > 0 and proved that lim c→0 ∇u c , where S is the best Sobolev constant.In the above results, the number p 0 := 2 + 4 N , called the L 2 -critical exponent, is crucial for the existence and asymptotic behavior of normalized solutions of (1.7).However, it is showed in [30,31] that when b = 0, In particular, when N = 3, the L 2 -critical exponent for the problem (1.5) is given by In [15], Li, Luo and Yang consider the existence and multiplicity of normalized solutions of (1.5) when N = 3 and prove that if 2 < q < 10 3 and 14 3 < p < 6, then for small µ > 0, E µ | Sc has a local minimizer at a negative energy level m(c, µ) < 0, and has a second critical point of mountain pass type at a positive energy level σ(c, µ) > 0. If 2 < q < 10 3 < p = 6, then for small µ > 0 a ground state solution is obtained.If 14  3 < q < p ≤ 6, then for any µ > 0, a critical point of mountain pass type is also obtained.Furthermore, as the parameter µ → 0 + , the asymptotic behavior of energy m(c, µ) and the normalized solution is also investigated.To our best knowledge, the existence of normalized solutions to (1.5) with 10  3 < q < p < 14   3   is still unknown.When µ = 0, then the equation (P λ ) reduces to the following Kirchhoff equation with a homogeneous nonlinearity Amongst other things, Li and Ye [31] studied the existence and concentration behavior of minimizers for (1.10) and obtained precise asymptotic behavior of normalized solutions to (1.10) with 2 < p < 2 + 8 N as c → +∞.For 2 < p < 2 + 4 N , Zeng and Zhang [33] proved the existence and uniqueness of the minimizer to the minimization problem E 0 (c) := inf u∈Sc E 0 (u) for any c > 0, while for 2 + 4 N ≤ p < 2 + 8 N the authors proved that there exists a threshold mass c * > 0 such that for any c ∈ (0, c * ) there is no minimizer and for c > c * there is a unique minimizer.Moreover, a precise formula for the minimizer and the threshold value c * is given according to the mass c.In the case 2 < p < 2 * , Qi and Zou [23] obtain the exact number and expressions of the positive normalized solutions for (1.10) and then answer an open problem concerning the exact number of positive solutions to the Kirchhoff equation with fixed frequency.Recently, Qihan He et al. [9] studied the existence and asymptotic behavior of normalized solutions of (1.10) with |u| p−2 u replaced by a general subcritical nonlinearity g(u) of mass super-critical type.In particular, g(u) contains the nonlinearity in (1.5) with 2 + 8 N < q ≤ p < 2 * as a special case.Under some suitable assumptions, they obtain the existence of ground state normalized solutions for any given c > 0. After a detailed analysis via the blow up method, they also described the asymptotic behavior of these solutions as c → 0 + , as well as c → +∞.
If both b = 0 and µ = 0, then (P λ ) become a nonlocal equation with a non-homogeneous nonlinearity.It is much more challenging and interesting to investigate the existence and qualitative properties of solutions of (1.5).Some existence results concerning the normalized solutions of (1.5) have been obtained over the past few years, see [15,32] and reference therein for Kirchhoff equations with combined powers nonlinearity.However, less progress is made on the asymptotic behavior of these solutions whenever the associated parameter varies in a suitable range.The technique used in [23,31,33,34] for the asymptotic study of equation (1.10) is not applicable any more, and any explicit expression of normalized solution in terms of the mass c is not available for the nonlocal problem (P λ ).
Our main purpose in this paper is to study the effect of the nonlocal term in the case b = 0 on the existence, non-existence, multiplicity and properties of normalized solutions of (1.5) and to understand the role of the L 2 -critical exponent p b in the existence and asymptotic behavior of normalized solutions of (1.5) as the parameter c varies.As a direct consequence of our main results on the fixed frequency problem (P λ ) in this paper, we are able to obtain an explicit asymptotic expression of different norms of positive normalized solutions, and to give a complete description on the existence, multiplicity and precise asymptotic behavior of positive normalized solutions of (1.5).In particular, we prove that both p 0 = 10 3 and p b = 14 3 play a key role in the existence, multiplicity and the asymptotic behavior of normalized solutions of (1.5) if b = 0.
Organization of the paper.In Section 2, we state the main results in this paper.In Section 3, we give a proof of Theorem 2.1 for small λ > 0. Section 4 is devoted to the proof of Theorems 2.1 for large λ > 0. In Section 5, we prove Theorem 2.2, and in the last section, as an application of our main results, we present some results concerning the existence, non-existence, and exact number of normalized solutions of the associated mass constrained problem.
Basic notations.Throughout this paper, we assume N ≥ 3.
• For any q ∈ (2, 2 * ) where 2 * = 2N N −2 , we define . (1.11) • B r denotes the ball in R N with radius r > 0 and centered at the origin, |B r | and B c r denote its Lebesgue measure and its complement in R N , respectively.• As usual, C, c, etc., denote generic positive constants.

Main results
The existence of ground state solutions established in [18] for Kirchhoff equations with a general nonlinearity and in [29,Theorem 2.1] for Kirchhoff equations with critical nonlinearites applies to (P λ ) directly or after a suitable scaling.In this paper, we are interested in the asymptotic behavior of ground state solutions of (P λ ).Our main results are the following two theorems.In the first one, we consider (P λ ) with p = 2 * in dimensions N = 3, 4. In the second one, we consider (P λ ) in dimension N = 3.

Proof of Theorem 2.1 as
Then the equation (P λ ) reduces to The energy functional for (Q λ ) is defined by The formal limit equation for (Q λ ) as λ → 0 is given by The energy functional for (Q 0 ) is given by ) and v is the rescaling (3.1) of u.Then: The above lemma is easily proved and the details will be omitted.In particular, it follows from Lemma 3.1 (c) that the rescaling v of the ground state u of (P λ ) corresponds to a ground state of (Q λ ).
Lemma 3.2.The rescaled family of ground-sates and the Pohožaev identity Therefore, it follows that and hence To prove the boundedness of Let v 0 be the unique positive solution of the equation (Q 0 ), then by the Pohožaev's identity, we have which implies that for small λ > 0, there is a unique t λ > 0 such that If N = 3 and t λ > 1, then If N = 4 and t λ > 1, then Therefore, we obtain .
On the other hand, we have This yields the boundedness of ∇v λ 2 for small λ > 0 and completes the proof.
Then for small λ > 0, there holds In particular, we have The proof of Lemma 3.3 is similar to that of [20, Lemma 3.2] and is omitted.Next we obtain an estimation of the least energy.Lemma 3.4.Let N = 3 or 4, and u λ is a ground state solution of (P λ ), then ), (3.9) as λ → 0, where m 0 := inf v∈H 1 (R N )\{0} sup t≥0 J 0 (tv) is the ground state energy for (Q 0 ).
On the other hand, by Lemma 3.3, we have .
The proof is complete.
Proof.Let u λ be a ground state solution of (P λ ) and Then the Nehari and Pohožaev identities imply that From which, we conclude that ). (3.14) Therefore, we obtain ).
In a similar way, we can show that Thus, we obtain ), which together with Lemma 3.4 implies that ). Since it follows from (3.14) and (3.15) that ), ), from which (3.11) and (3.12) follows.The proof is complete.
Proof of Theorem 2.1 for small λ.Observe that v λ → v 0 in H 1 (R N ) with v 0 being the unique ground state solution of (Q 0 ).For small λ > 0, Theorem 2.1 follows from Lemmas 3.1-3.5 and the details will be omitted.
The formal limit equation for (R λ ) as λ → ∞ is given by the equation The corresponding functional is given by We denote their corresponding Nehari manifolds as follows: Then are well-defined and positive.
then the equation (R ∞ ) reduces to Moreover, satisfies the equation ) and the family of its rescalings where W 1 is the the Talenti function .
Furthermore, a direct computation shows that ) and v is the rescaling (4.1) of u.Then: In particular, if v λ is the rescaling (4.1) of the ground state u λ , then and hence v λ is the ground state of (R λ ).Moreover, v λ satisfies the Pohožaev's identity [4]: Define the Pohožaev manifold where Clearly, v λ ∈ P λ .Moreover, we have the following minimax characterizations for the least energy level m λ : In particular, we have m λ = J λ (v λ ) = sup t>0 J λ (tv λ ) = sup t>0 J λ ((v λ ) t ).
Proof.The Nehari identity and the Pohožaev identity So, to prove the boundedness of {v λ } in H 1 (R N ), it suffices to show that {v λ } is bounded in D 1,2 (R N ).It is easy to see that On the other hand, it follows from Lemma 4.3 below that m λ ≤ C < +∞ for large λ > 0, and hence {v λ } is bounded in D 1,2 (R N ) and H 1 (R N ).
Next we obtain an estimation of the least energy.Lemma 4.3.There exists a constant C = C(q) > 0 such that for all large λ > 0, if N = 3 and q > 4. (4.13) For 1, a straightforward computation shows that By Lemma 3.4, we find where t λ > 0 is the unique solution of the following equation 2) ).We also deduce , where t > 0 is given by Hence, we have ).Therefore, we obtain Therefore, we get Thus, we obtain On the other hand, we have Then h(ρ) take its maximum value ϕ(ρ ) at the unique point ρ > 0, and Then we obtain For the rest of the proof we consider separately the cases N = 4 and N = 3.
which finished the proof in the case N = 3.
Proof.Arguing as in the proof of Lemma 4.3, it is easy to show that which together with Lemma 4.3 yields the desired conclusion.

4.2.
Proof of Theorem 2.1 for large λ.We recall the P.-L.Lions' concentration-compactness lemma, which is at the core of our proof of Theorem 2.1 as λ → ∞.
Lemma 4.6.If N = 4 or N = 3 then there exists ξ λ ∈ (0, +∞) such that ξ λ → 0 and Proof.Note that v λ is a positive radially symmetric function, and by Lemma 4.2, and Observe that Therefore, {v λ } is a (P S) sequence for J ∞ at level m ∞ .By Lemma 4.5 and an argument similar to that in [28], it is standard to show that there exists ζ where ṽλ λ → 0 as λ → ∞, and v (j) are nontrivial solutions of the equation −(a + bA 2 )∆v = v 2 * −1 .Moreover, we have and where For any solution v of the equation −(a + bA 2 )∆v = v 2 * −1 , we have Therefore, we obtain If N = 4 or 3 then by (4.12) and Fatou's lemma we have Note that v ∞ / ∈ L 2 (R N ) whenever v ∞ = 0, therefore, v ∞ = 0 and hence k = 1.Thus, we obtain J ∞ (v (1) ) = m ∞ and hence v (1) = v ρ for some ρ ∈ (0, +∞).Therefore, we conclude that where v 1 is given by (4.7) and ξ λ := ρζ We perform an additional rescaling where ξ λ ∈ (0, +∞) is given in Lemma 4.6.This rescaling transforms (Q λ ) into an equivalent equation The corresponding energy functional is given by Jλ (w It is straightforward to verify the following. Lemma 4.7.Let λ > 0, v is the rescaling of u ∈ H 1 (R N ) and w is the rescaling of v given in (4.1) and (4.24), respectively.Then: where the v λ is a ground-state of (R λ ).Then it follows from Lemma 4.7(c) that w λ is a ground state of ( Rλ ).By Lemma 4.6 we conclude that Note that the corresponding Nehari and Pohožaev identities read as follows We conclude that Thus, we obtain To control the norm w λ 2 from below, we give the following estimate: Lemma 4.8.There exists a constant C > 0 such that ) Proof.It is easy to see that wλ (x) = w λ ( √ λ x) satisfies the following Arguing as in the proof in [21, Lemma 4.8], we show that Therefore, we obtain The proof is complete.
To prove our main result, the key point is to show the boundedness of w λ q .
Similarly, we have Therefore, we obtain which together with (4.35) implies that Finally, by (4.27), (4.35) and Lemma 4.11, we obtain Statements on u λ follow from the corresponding results on v λ and w λ .This completes the proof of Theorem 2.1 for large λ.

Proof of Theorem 2.2
In this section, we aways assume N = 3.We divide this section into three subsections, in which we consider the cases (i) p = q, (ii) q < p, λ > 0 large, and (iii) q < p, λ > 0 small, respectively.If 2 < q ≤ p < 2 * , then the arguments used in Section 3 and Section 4 do not work any more, so we try to transform the nonlocal equation (P λ ) into the local equation (1.3) by using a suitable rescaling which is dependent of the ground state solution.Unfortunately, if b = 0, such a rescaling does not neceessarily transforms a ground state solution into a ground state solution of a new equation, which prevents us from deriving a precise energy estimate of the ground state.
5.1.The case p = q.Let W ∈ H 1 (R 3 ) is the unique positive solution of the equation and S p = W p−2 p .Let u λ be a ground state solution of (P λ ), then for any λ > 0, there holds where λ = a if b = 0, and if b = 0, then ) , as λ → 0, as λ → 0.
For b = 0, we have 5.2.The case q < p and λ > 0 is sufficiently large.Let u λ be a ground state solution of (P λ ), and Then w = w λ satisies The corresponding functional is given by Observe that it follows that (5.5) where On the other hand, assume w λ is a critical point of J λ and λ is given by (5.4), then is a critical point of I λ .These observation reduces the problem of finding critical point of I λ to the corresponding problem of finding critical point of J λ .For general λ > 0, the ground state solutions of (5.3) should not be unique.But for large λ > 0, this is not the case.Arguing as in [13, Theorem 5.1] (see also [1]), by the implicit function theorem, we can show that for large λ > 0, (5.3) admits a unique positive ground state solution, therefore, (P λ ) has only one ground state solution for large λ > 0. This also yields that w λ is a ground state solution of (5.3).
We mention that when 2 < q < 10 3 , 14 3 < p < 6, a nontrivial variation of M (λ) can also be observed in Figure 6, which affects the existence, non-existence and multiplicity of normalized solutions of (1.5).This special behavior of M (λ) is mainly caused by the combined nonlinearity and have been observed in nonlinear Schrödinger equations and Kirchhoff equations with general subcritical nonlinearity [13,9].We also mention that this type of behavior of M (λ) does not appear in the Kirchhoff equations with a pure power nonlinearity [23].Typically, the asymptotic behavior of M (λ) depicted in Figure 5 is mainly caused by the appearance of the nonlocal term b R N |∇u| 2 , which has been reported by Qi and Zou [23] as a new phenomenon for Kirchhoff equation with a pure power nonlinearity.While the asymptotic behaviors of M (λ) depicted in Figure 7 and Figure 8 are mainly caused by the combined effect of the nonlocal term and the combined nonlinearity, which have not been reported before in the literature.
Besides, the value of b > 0 has also an effect on the existence, non-existence and the number of normalized solutions of (1.5), which can be seen from Figure 8 and the following diagrams.

1 p− 2 W
(0), and a direct computation shows that for b = 0,