NORMALIZED SOLUTIONS OF HIGHER-ORDER SCHR¨ODINGER EQUATIONS

. In this paper, we consider the existence of non-trivial solutions for the following equation where H 0 J is the higher-order Schr¨odinger operator with J ∈ N , 2 < p < 4 J +63 , and λ ∈ R is a parameter. Let E ( u ) be the corresponding variational functional of problem (1). We look for solutions of equation (1) by ﬁnding minimizers of the minimization problem We show that problem (1) admits at least a solution provided that in the case J being odd, 2 < p < 3 and ρ > 0 small or 2 + J < p < 4 J +63 and ρ > 0 large; and for the case J being even, 3 < p < 4 J +63 and ρ > 0 small.

1. Introduction. In this paper, we investigate the higher-order Schrödinger equation where H 0J is the higher-order kinetic energy operator given by with J ∈ N. The constant = h 2π denotes the reduced Planck constant, c is the speed of light in the vacuum, and α(j) = (2j − 2)! j!(j − 1)!2 2j−1 for j ≥ 1 with α(0) = −1. The operator H 0J can be regarded as the finite approximation of the operator while the operator H 0∞ stems from the study of the pseudo-relativistic operator √ −c 2 2 ∆ + m 2 c 4 . Indeed, to describe the motion of a fast moving free particle, one has to take into account the effect of the speed of light, then from the Einstein's mass-energy equivalence we obtain the kinetic energy E = p 2 c 2 + m 2 c 4 (3) of a free particle of mass m and momentum p. Applying the correspondence principle we obtain the pseudo-relativistic equation with the non-local pseudo-differential operator √ −c 2 2 ∆ + m 2 c 4 , which was studied in [17]. Further study of nonlinear problems with the operator √ −c 2 2 ∆ + m 2 c 4 can be found in [5,16,21] etc.
Historically, in order to avoid the non-local operator, one squares both sides of (3) and obtains E 2 = p 2 c 2 + m 2 c 4 , which yields the Klein-Gordon equation Since (5) is of second order derivative in time, it cannot be interpreted as a Schrödinger type equation for a quantum state. Another way in this consideration is so called the Dirac equation, formulated by Dirac [7], that gave the fundamental relativistic equation of first order in time. On the other hand, in [4], using the Taylor series expansion of (3) in p, and by the correspondence principle (4), Carles and Moulay obtain the Schödinger form equation The convergence of finite approximation H 0J to H 0∞ was considered in [4]. It is proved in [4] that equation (6) holds if 2v 2 < c 2 . Schrödinger type equations involving a higher-order Schrödinger operator and converging towards the semirelativistic bound-state equation is known as the spinless Salpeter equation, it was studied in [11,12], see also [6,10,13,14] and references therein. The Cauchy problem of the higher-order Schrödinger equations without potential, i.e., for free particles, is considered in [4,15]. It is treated in [4] the case of bounded potentials, e.g., particles in finite potential wells, and of linear potentials, that is, neutrons in free fall in the gravity field and electrons accelerated by an electric field. While in [3], it is studied the Hartree-Fock equations with harmonic-oscillator and Coulomb potentials. Moreover, the higher-order Schrödinger operator with quasi-periodic potentials in two dimensions is discussed in [14]. We remark that for J = 1, H 0J becomes the regular Schrödinger operator ∆.
If J = 2, we get Such an operator was considered in [19,20] for the Woods-Saxon problem. It is obvious that operators H 01 and H 02 enjoy different features.
In this paper, we restrict to the case J > 1. For notational simplicity, in the sequel we take without lose of generality that the constants equal to 1, that is, Now, we consider the existence of solutions for equation (2) under the constraint with 2 < p < 4J+6 3 . Since the operator H 0J emerges different properties by J being even or odd, it is necessary to treat the operator H 0J separately.
The natural way to study the problem is to look for critical points of the functional if J = 2k + 1 for k = 0, 1, 2, · · · ; or E even (u) = 1 2 if J = 2k for k = 1, 2, · · · under the constraint (7). Denote by H J (R 3 ) the usual Sobolev spaces. Then, any critical point of both E(u) = E odd (u) and E even (u) constrained to S ρ := u ∈ H J (R 3 ) : u L 2 (R 3 ) = ρ (9) corresponds to a solution of (2). By a solution of (2) we mean a couple (λ ρ , u ρ ) ∈ R × H J (R 3 ), where λ ρ is the Lagrange multiplier associated with the critical point u ρ on S ρ . However, the functional E even (u) is not bounded from blow on S ρ . Hence, we define E even (u) = −E even (u), that is, and thus we set E(u) =Ẽ even (u) if J is even. Our main result is as follows.
Since our problem (2) is setting in R 3 , the main difficulty for variational problems is the lack of compactness. According to the concentration-compactness principle [18], the loss of the compactness occurs if vanishing and dichotomy happen. Correspondingly, for a minimizing sequence {u n } ⊂ S ρ of E(u), we have either (i) u n 0 or (ii) u n ū = 0 and 0 < ū L 2 (R 3 ) < ρ. In order to show the relatively compactness of minimizing sequence {u n } ⊂ S ρ of E, the necessary and sufficient condition is the following subadditivity inequality for all 0 < µ < ρ. This inequality allows us to rule out vanishing and dichotomy. The classical approach to prove the subadditivity inequality (11) is to show the function s → s −2 E s is monotone decreasing. However, this assertion is not easy to prove. In [1,2], Bellazzini and Siciliano find a way to recover the monotone decreasing condition, and applied it to the Schrödinger-Poisson equation.
We will deal with the problem E ρ by the concentration-compactness principle. As the functional E(u) has different features for J being odd and even, we distinguish two cases to discuss. In the case that J is odd, we may show the subadditivity inequality for E odd ρ if 2+J < p < 4J+6 3 by the standard scaling arguments. However, in the case J being even, the standard scaling arguments do not permit to show that the subadditivity inequality (11). Hence, we will use the techniques introduced in [2]. This paper is organized as follows. In Section 2, we present some useful inequalities and prove a lemma. Sections 3 and 4 are devoted to prove our main results for J being odd and even, respectively .

Preliminaries.
In this section, we present some useful inequalities and facts for future references.
Denote by H m (R N ) the Sobolev space endowed with the norm We recall the following useful inequalities: and a natural number m. Suppose also that a real number α and a natural number j are such that Then every function u : R n → R that lies in L q (R n ) with m-th derivative in L r (R n ) also has j-th derivative in L p (R n ); Moreover, there exists a constant C depending only on m, n, j, q, r and α such that NORMALIZED SOLUTIONS OF HIGHER-ORDER SCHRÖDINGER EQUATIONS 451 (II) (Interpolation inequality ) For j, p ≥ 0, denote Now, we present an abstract result on a constrained minimization problem in Sobolev space H J (R 3 ). Let us consider the following problem where Under suitable assumptions on T , we show as [1,2] that any minimizing sequence of problem (14) actually converges strongly.
where α n = ρ 2 − µ 2 / u n −ū L 2 (R 3 ) and for any 0 < µ < ρ. Thenū ∈ S ρ . Moveover if, as n, m → ∞, Proof. We only consider the case that J being odd since the same argument can be carried out for the case J being even. We argue by contradiction. Suppose on the contrary that µ < ρ. Since u n −ū 0, we have Since {u n } is a minimizing sequence, and by (15), we derive Using (16) and (20), we find , which is in contradiction with (17). This implies that ū L 2 (R 3 ) = ρ.
In order to prove that u n →ū in H J (R 3 ), we may assume, by Ekeland variational principle [9], that {u n } is a Palais-Smale sequence for functional E. Sinceū ∈ B ρ , then u n −ū L 2 (R 3 ) = o(1) and thus it remains to show that |u n −ū| J,2 = o(1).
By assumptions, there exists a sequence {λ n } ⊂ R such that for functional E, where ·, · denotes the duality pairing. It follows that is bounded. From this and assumption (19) we see that the sequence {λ n } is bounded, and thus up to a subsequence there exits λ ∈ R such that λ n → λ. We now have (1), λ n → λ, by (18) and interpolation inequality (13), we obtain that {u n } is a Cauchy sequence in H J (R 3 ). Hence u n −ū H J (R 3 ) → 0.

NORMALIZED SOLUTIONS OF HIGHER-ORDER SCHRÖDINGER EQUATIONS 453
We will apply Lemma 2.1 to the functional E(u) with in the proof of Theorem 1.1. The next lemma shows that the functional E(u) is bounded from below on S ρ .
Proof. By the Galiardo-Nirenberg inequality, we have where C ρ and C ρ are positive constants which depending on ρ. Since p < 4J+6 3 , it results αp < 2, which implies E odd (u) is bounded from below and coercive on S ρ .
On the other hand, by the interpolation inequality, we have It follows that E even (u) is bounded from below and coercive on S ρ . The proof is complete.
3. The existence for J being odd. This section is devoted to prove the existence result when J is odd, that is, Theorem 1.1 (a) and (b). As the functional E odd ρ behaves differently for 2 < p < 2 + J and 2 + J < p < 4J+6 3 , we distinguish two cases (i) 2 < p < 2 + J and (ii) 2 + J < p < 4J+6 3 to discuss. We commence with establishing the subadditivity inequality.

NORMALIZED SOLUTIONS OF HIGHER-ORDER SCHRÖDINGER EQUATIONS 455
we have (26) Since µ < ρ, we distinguish the following cases: (1) µ < ρ 2 − µ 2 ; For the first case, by (26), we have For the second case, we choose θ = √ 2 in (26) and the assertion follows. For the third case, by (26), we have Now we are in position to prove Theorem 1.1 (a) and (b).
Proof of Theorem 1.1 (a) and (b). Let {u n } ⊂ S ρ be a minimizing sequence of E odd ρ . Notice that for any sequence y n ∈ R 3 , u n (·+y n ) is still a minimizing sequence for E odd ρ . By Lemma 2.1, it suffices to show that there is a sequence y n ∈ R 3 such that the weak limit of u n (· + y n ) belongs to S ρ , and then the convergence becomes strongly in H J (R 3 ). Thus, the assertion follows. To this purpose, we now rule out the vanishing.
4. The existence for J being even. This section is devoted to prove the existence result when J is even, that is, Theorem 1.1 (c). In this case, the standard scaling arguments do not permit to show that the subadditivity inequality holds for 0 < µ < ρ. Hence, the possibility of dichotomy for an arbitrary minimizing sequence cannot be excluded. By techniques introduced in [2], we are able to prove (27) holds for J being even at least for small value of ρ. Hence, the compactness of minimizing sequences of E even ρ retains up to translations. The main idea is to show that the function µ → E even µ /µ 2 is monotone decreasing, which yields (27). Indeed, if the function µ → E even µ /µ 2 is monotone decreasing for µ ∈ (0, ρ), we have which implies for any µ ∈ (0, ρ). Now, we recall some definitions in [2].
is differentiable and Θ gu (1) ≡ 0. Denote by G u the set of the scaling paths of u.
In the application, it is relevant to consider the family of scaling paths of u parameterized with β ∈ R given by Observe that all paths of this family have the associated function Θ gu (θ) = θ 2 .
Finally, let u ≡ 0 be fixed and g u ∈ G u . We say that the scaling path g u is admissible for the functional E even if g u is a differentiable function.
The following result allow us to find a minimizer of E even ρ . Proposition 4.1. Assume that for every ρ > 0, all the minimizing sequences {u n } for E even has a weak limit, up to translations, different from zero. Assume for all 0 < µ < ρ (28) and the following conditions s → E even s is continuous, Proof. The proof is similar to that of Theorem 2.1 in [2], so we omit it here.
Next, we need the following estimate.
Now we are in position to prove Theorem 1.1 (c).
Proof of Theorem 1.1 (c). We will verify that the hypotheses of Proposition 4.1 are fulfilled, the conclusion then follows from Proposition 4.1.
Next, we prove that the function s → E even s satisfies (30) and (31). We claim that lim n→∞ E even ρn = E even ρ if ρ n → ρ. In fact, for every n ∈ N, let w n ∈ S ρn be such that On the other hand, given a minimizing sequence {v n } ⊂ S ρ for E even ρ , we have which, together with (33), yields lim n→∞ E even ρn = E even ρ . The claim follows. Now, we show that We argue indirectly. Suppose on the contrary that lim ρ→0 E even ρ ρ 2 = −2c < 0 for some c ∈ (0, 1/2). Therefore, E even Then, u ρ n satisfies J j=1 ∆ j u ρ n + |u ρ n | p−2 u ρ n − λ ρ n u ρ n = σ ρ n → 0 (36) as n → ∞. By (36) and Galiardo-Nirenberg inequality, we have for some constants C 1 , C 2 > 0 and α given by (21). Hence, fixing ρ > 0 small, we have λ ρ n ≤ −c, since (1 − α)p > 2. By (36) and Sobolev embedding, where C is a positive constant and o(1) → 0 as n → ∞. On the other hand, by interpolation inequality (13), we can show that since u ρ n L 2 (R 3 ) = ρ is small. Therefore, we deduce from (37) and (38) that there existsc > 0 such that u ρ n 2 H J (R 3 ) ≥c > 0 for n large enough. However, by (23), we have if ρ is small, which contradicts to (35). Hence, (34) holds true. Finally, we show the strict subadditivity inequality holds true, which implies the strong convergence of minimizing sequences of E even ρ . By Proposition 4.1, it is enough to verify that the functional E even satisfies (32) for small ρ > 0.
We argue by contradiction. Were it not the case, there would exist a sequence {u n } ⊂ M (ρ) with ρ ≥ u n L 2 (R 3 ) = ρ n → 0 such that for all β ∈ R = 0 Using (39), we have |u n | p dx = 0.
Since p > 2, u n L p (R 3 ) = 0, which yields a contradiction. The claim is true and the proof is complete.