Normalized Solutions of Nonautonomous Kirchhoff Equations: Sub- and Super-critical Cases

In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation {− (a + b ∫ R3 |∇u|2dx) u − λu = K (x) f (u), x ∈ R3; u ∈ H1(R3), where a, b > 0, λ is unknown and appears as a Lagrange multiplier, K ∈ C(R3,R+) with 0 < lim|y|→∞ K (y) ≤ infR3 K , and f ∈ C(R,R) satisfies general L2-supercritical or L2-subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f . This paper extends to the nonautonomous case previous results on prescribed L2-norm solutions of Kirchhoff problems.


Introduction
This paper deals with the existence of normalized solutions to the following nonautonomous Kirchhoff-type equation: where a, b are positive real numbers, λ is unknown and will appear as a Lagrange multiplier, K ∈ C(R 3 , R + ) and f ∈ C(R, R). This equation is related to the stationary analogue of the Kirchhoff equation 2) The Kirchhoff equation has been introduced for the first time in 1883 by Kirchhoff [16] in dimension 1, without forcing term and with Dirichlet boundary conditions, in order to describe the transversal free vibrations of a clamped string in which the dependence of the tension on the deformation cannot be neglected. This is a quasilinear partial differential equation, namely the nonlinear part of the equation contains as many derivatives as the linear differential operator. The Kirchhoff equation is an extension of the classical D'Alembert wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. We refer to [1,5,6,12] for the physical background on Kirchhoff's model.
From the mathematical point of view, problem (1.1) is nonlocal since the appearance of the term R 3 |∇u| 2 dx indicates that (1.1) is not a pointwise identity. This kind of problem has been paid much attention after the pioneering work of Lions [19], in which an abstract functional analysis framework was introduced.
If λ ∈ R is a fixed parameter or even in the presence of an additional external and fixed potential V (x), the existence of solutions of problem (1.1) has been intensively studied during the last decade; see, for example, [2,7,13,17,18,[20][21][22]24,32] and the references therein. In this case, solutions can be obtained as critical points of the corresponding energy functional, but without any information on the L 2 -norm of the solutions.
Nowadays, since physicists are interested in normalized solutions, mathematical researchers began to focus on solutions having a prescribed L 2 -norm, that is, solutions which satisfy u 2 2 = c > 0 for a priori given c. To the best of our knowledge, the study of solutions with prescribed norm was initiated by Jeanjean [14] in the framework of semilinear elliptic equations. We also refer to Bellazzini et al. [4] and Cingolani and Jeanjean [11] for normalized solutions of the Schrödinger-Poisson system. In the present paper, we are interested in the existence of solutions with L 2 -prescribed norm and their qualitative properties in the framework of nonlocal Kirchhoff problems. Our analysis includes both the L 2 -supercritical case and the L 2 -subcritical growth.
Solutions of problem (1.1) with u 2 2 = c > 0 can be obtained by looking for critical points of the following functional where F(u) = u 0 f (t)dt. In this case, the parameter λ ∈ R cannot be fixed but instead it appears as a Lagrange multiplier, and each critical point u c ∈ S c of I | S c , corresponds to a Lagrange multiplier λ c ∈ R such that (u c , λ c ) solves (weakly) problem (1.1). In particular, if u c ∈ S c is a solution of the constrained minimization problem σ (c) := inf u∈S c I (u), (1.5) then there exists λ c ∈ R such that I (u c ) = λ c u c , that is, (u c , λ c ) is a solution of problem (1.1).
To the best of our knowledge, solutions of problem (1.1) having a prescribed L 2norm have been studied only if K (x) ≡ 1; see, e.g., [28][29][30][31]33]. Let us introduce and review the few known results in this respect. Ye [29] studied the existence and non-existence of normalized solutions to the special form of (1.1): for p ∈ (2, 6), and showed that p = 14 3 is a L 2 -critical exponent for problem (1.6), that is, for any given c > 0, σ (c) ∈ (−∞, 0], if p ∈ (2, 14 3 ); σ (c) = −∞, if p ∈ ( 14 3 , 6). (1.7) More precisely, for any p ∈ (2, 14 3 ), Ye [29] obtained the sharp existence of global constraint minimizers for (1.6) by solving the minimization problem (1.5). If p ∈ ( 10 3 , 14 3 ), Ye [29] found a local minimizer which is also a critical point of I | S c by constructing a geometry of local minima for I . For the case p ∈ ( 14 3 , 6), since the minimization problem (1.5) is not available due to (1.7), to look for a critical point of I | S c , Ye [29] took a minimum on a suitable submanifold 8) and showed that M c is a natural constraint of I | S c by using the Lagrange multiplier method, where and u t ∈ S c for all t > 0 if u ∈ S c . Note that this method relies heavily on the powertype nonlinearity f (u) = |u| p−2 u with p ∈ ( 14 3 , 6). Recently, in [28], Xie and Chen generalized this special case to general nonlinearities f satisfying lim |t|→∞ F(t) |t| 14/3 = +∞. Using some ideas developed in [24,29], Xie and Chen [28] proved the existence of normalized solutions for problem (1.1) with K (x) ≡ 1 under additional growth and monotonicity conditions. For the L 2 -critical case, Ye [30] proved that problem (1.6) with p = 14 3 has a solution (u c , λ c ) which satisfies u c 2 2 = c > c * for some c * > 0. We also recall that Ye [31] also analyzed the concentration behavior of solutions. If p ∈ (2, 14 3 ), by using a different method, Zeng and Zhang [33] proved the existence and uniqueness of normalized solutions for problem (1.6). Additionally, it was considered in [29][30][31]33] the existence of normalized solutions for one-dimensional and twodimensional autonomous Kirchhoff type equations with power-type nonlinearity.
Let us also emphasize that all of the strategies used in [28][29][30][31]33] only work for autonomous problems, and fail to adapt directly to problem (1.1) with non-constant potential K (x). To the best of our knowledge, there are no results dealing with the non-autonomous abstract setting. The main purpose of this paper is to extend and complement the corresponding existence results in [29] to problem (1.1) in the presence of the variable potential K (x).

Main Results
Motivated by the above works, we first consider the L 2 -supercritical case, and establish the existence of a critical point of I on S c by considering the constrained minimization problem m(c) := inf where the definition of M c is given by (1.8). To this end, we introduce the following assumptions: 14 3 ) such that lim |t|→0 Our first result establishes the following qualitative property.
where the definitions of M c and u t are given by (1.8) and (1.9).
Next, in the L 2 -subcritical case, we find a global minimizer and a local minimizer of I which are critical points of I | S c by solving the minimization problem (1.5) and constructing a geometry of local minima for I (see Lemma 4.6), respectively. To this end, in addition to (K1), we introduce the following assumptions: 10 3 ) such that lim |t|→0 We have the following statement.

(2.3)
Moreover, for the above critical point u c , there is a Lagrange multiplier λ c ∈ R such that (u c , λ c ) is a solution of problem (1.1).

Remark 2.3
Theorems 2.1 and 2.2 make a substantial improvement and extension to the main results in [29]. In particular, if K (x) ≡ 1, the conclusion of Theorem 2. Compared with the previous works, we have to overcome the essential difficulties that the variable potential K (x) gives rise when searching for normalized solutions of problem (1.1). These difficulties enforce the implementation of new ideas and techniques for the proof of Theorems 2.1 and 2.2. Let us point them out in more detail.
For the L 2 -supercritical case, some useful remarks are stated in what follows.
• When K (x) ≡ K ∞ , Ye [29] showed that c → m(c) is strictly decreasing on (0, +∞) (2.4) by using the translation invariance of I and the homogeneity of f . Then Ye can exclude the vanishing and the dichotomy cases of the minimizing sequence {u n } for m(c) = inf u∈M c I (u) in applying the concentration-compactness principle. However, the approach used in [29] is valid only for autonomous equations and it does not work any more for (1.1) with K = constant and more general f , see Remark 3.10 for more details. Unlike [29], by establishing some new inequalities, we prove that m(c) is nonincreasing and m(c) > m(c) for anyc > c provided m(c) is attained. To bypass the difficulty caused by the lack of compactness of Sobolev embedding H 1 (R 3 ) → L s (R 3 ) for 2 ≤ s < 6, we compare the constrained minimum m(c) with the one of the "limit equation" (that is, problem (1.1) with K (x) = K ∞ ), and by using some subtle analysis we prove that u n →ū in • To verify that M c is a natural constraint on S c , we use a combination of the deformation lemma, some new inequalities and an intermediary theorem for continuous functions, other than the mountain pass theorem on S c and the Lagrange multiplier method used in [28,29], respectively. For the L 2 -subcritical case, some remarks are as follows. • The key step to prove that σ (c) = inf S c I is achieved is to obtain the subadditivity inequality σ (c) < σ (α) + σ (c − α), ∀ 0 < α < c. (2.5) To this end, Ye in [29] used the scaling t → u(t −2/3 x). But this kind of scaling is not suitable in our case, excepting the case when K (x) is a positive constant. Instead, we present another scaling t → t 1/2 u(x/t), and we succeed to prove that (2.5) still holds under assumptions of Theorem 2.2. • Compared with [29], it is more complicated to find a local minimizer of I on S c which is a critical point of I | S c because K (x) is variable and f is nonhomogeneous. For this purpose, we make some improvements of the method used in [29] and employ some subtle analysis in the proofs.
When K ∈ C(R 3 , R + ) is bounded, f satisfies (F1) and (F2) (or (F4)), we deduce by a standard argument that I ∈ C 1 (H 1 (R 3 ), R). Let us define the "limit equation" associated to problem (1.1) by Remark that all above conclusions on problem (1.1) in this paper are also true for the limit equation Let a > 0 and b > 0 be fixed. Throughout this paper we make use of the following notations: • H 1 (R 3 ) denotes the usual Sobolev space equipped with the inner product and norm • For any x ∈ R 3 and r > 0, B r (x) := {y ∈ R 3 : |y − x| < r }; . . denote positive constants possibly different in different places.

First Existence Result
In this section, we give the proof of Theorem 2.1.

Lemma 3.3 Assume that hypotheses
By the scaling (1.9), we have and , it follows from (3.9) and (3.10) that and Inspired by [9,10], we prove the following lemma.
From Lemma 3.4, we have the following corollary. In what follows, the definitions of M c and u t are given by (1.8) and (1.9).
From (3.6) and (3.20), we derive that for any s ∈ R, which concludes the proof of (i).

This contradiction shows that m(c) ≤ m ∞ (c).
Similarly to [  This relation together with u n 2 2 = c, implies that {u n } is bounded in H 1 (R 3 ). Passing to a subsequence, we have u n ū in H 1 (R 3 ). Then u n →ū in L s loc (R 3 ) for 2 ≤ s < 6 and u n →ū a.e. in R 3 . There are two possible cases: i)ū = 0 and ii)ū = 0.

Second Existence Result and Qualitative Properties of Solutions
In this section, we give the proof of Theorem 2.2. Proof (i) Using (F4), for any ε > 0, there exists C ε > 0 such that

Global Minimizers on the Constraint S c
2) This relation together with 0 < 3( p − 2)/2 < 4, shows that I is bounded from below on S c for any c > 0, that is, σ (c) is well defined. Noting that u t ∈ S c for all u ∈ S c , from (3.9) and (4.1), we deduce that I (u t ) → 0 as t → 0, and so σ (c) ≤ 0 for any c > 0.
Noting that Lemma 4.1 implies we have is well-defined.    On the other hand, given a minimization sequence {v n } ⊂ S c for I , we have which, jointly to (4.14), gives lim n→∞ σ (c n ) = σ (c).
(ii) Note that (1.3) and (4.6) lead to 15) where the definition of u t is given by (4.5).
Let {u n } ⊂ S c be such that I (u n ) → σ (c) for any c > 0. Since (u n ) t 2 2 = t 4 u n 2 2 = t 4 c for all t > 0, it follows from (4.15) that Let {u n } ⊂ S c be such that I (u n ) → σ (c) for any c > c * . We claim that exists a constant ρ 0 > 0 such that lim inf n→∞ ∇u n 2 > ρ 0 .
(ii) In view of Lemma 4.1 (ii), we have σ (c) < 0 for any c > 0. Let {u n } ⊂ S c be such that I (u n ) → σ (c) for any c > 0. Then (4.2) implies that {u n } is bounded in H 1 (R 3 ). Thus, we can assume that for someū ∈ H 1 (R 3 ) and up to a subsequence, u n ū in H 1 (R 3 ). Here, we distinguish two cases: a)ū = 0 and b)ū = 0.
This shows thatû is a minimizer of σ (c) for any c > 0.

Local Minimizers on the Constraint S c
In this subsection, we shall look for a local minimizer of I on the constraint S c , which is a critical point of I | S c . For k > 0, set For any 0 < c < c * and u ∈ S c , it follows from (1.3), (3.23) and (4.31) that (4.33) Since 3 p − 10 > 0 and ε is arbitrary, there exists k 0 > 0 independent of c such that In view of Lemma 4.1 (i), we havē Then it follows from (4.37) that which completes the proof. (4.43) such that, up to a subsequence, u n ū c in H 1 (R 3 ), u n →ū c in L s loc (R 3 ) for 2 ≤ s < 6 and u n →ū c a.e. in R 3 . For brevity, we denoteū c byū. We complete our proof in two steps as follows.