Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 29, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.29 NORMALIZED GROUND STATE OF A MIXED DISPERSION NONLINEAR SCHRÖDINGER EQUATION WITH COMBINED POWER-TYPE NONLINEARITIES ZHOUJI MA, XIAOJUN CHANG, ZHAOSHENG FENG Abstract. We study the existence of normalized ground state solutions to a mixed dispersion fourth-order nonlinear Schrödinger equation with combined power-type nonlinearities. By analyzing the subadditivity of the ground state energy with respect to the prescribed mass, we employ a constrained mini- mization method to establish the existence of ground state that corresponds to a local minimum of the associated functional. Under certain conditions, by studying the monotonicity of ground state energy as the mass varies, we apply the constrained minimization arguments on the Nehari-Pohozaev manifold to prove the existence of normalized ground state solutions. 1. Introduction and main results Consider the mixed dispersion nonlinear Schrödinger equation with combined power-type nonlinearities i∂tψ − ϵ∆2ψ + γ∆ψ + µ|ψ|q−2ψ + |ψ|p−2ψ = 0, (1.1) where N ≥ 1, µ ≥ 0, ϵ ≥ 0, γ ∈ R, ψ ∈ R × RN → C and 2 < q < p ≤ 4∗. Note that equation (1.1) becomes the well-known Schrödinger equation when ϵ = 0 and γ = 1. This equation has been extensively studied as a partial differential equation, presenting various mathematical challenges from the perspective of mathematical physics [4, 6]. Over the past decades, a lot of attention has been paid to normalized solutions of the nonlinear Schrödinger equation with both pure and mixed nonlin- earities [1, 7, 10, 11, 12, 13, 17, 18, 19, 22, 23, 26, 34, 35, 38] and the references therein. For the specific case µ = 0, when 2 < p < 2 + 4 N , all solutions to (1.1) with ϵ = 0 exist globally, and the associated standing waves are orbitally stable. However, for p ≥ 2 + 4 N , the solutions to equation (1.1) can exhibit singularity within a finite time. To address regularization and stabilization of these solutions, Karpman-Shagalov [21, 20] proposed the inclusion of a small fourth-order disper- sion term ϵ∥∆u∥22 in the model. Through a combination of stability analysis and numerical simulations, they demonstrated the stable outcomes for 2 < p < 2 + 8 N , while noting the instability phenomena for p ≥ 2 + 8 N . Consequently, p = 2 + 8 N 2020 Mathematics Subject Classification. 35Q55, 31B30, 35J30. Key words and phrases. Normalized solutions; Schrödinger equation; Lagrange multiplier; ground states; Nehari-Pohozaev manifold. ©2024. This work is licensed under a CC BY 4.0 license. Submitted November 18, 2023. Published April 1, 2024. 1 2 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 appears as a new mass critical exponent. Despite the significance of the mixed dis- persion fourth-order nonlinear Schrödinger equation in physical contexts, it remains inadequately understood, as addressed in [4, 8, 15, 29, 30, 32]. In this article, we are concerned with equation (1.1) and its standing waves solutions of the form ψ(t, x) = eiωtu(x), where ω ∈ R is a Lagrange multiplier and u(x) satisfies ϵ∆2u− γ∆u+ ωu− µ|u|q−2u− |u|p−2u = 0 in RN . (1.2) When we consider solutions to (1.2), a possible choice is to consider a fixed value ω ∈ R and search for solutions as the critical points of the action functional Aω,µ(u) = ϵ 2 ∥∆u∥22 + γ 2 ∥∇u∥22 + ω 2 ∥u∥22 − µ q ∥u∥qq − 1 p ∥u∥pp. In this case, we focus on the existence of minimal action solutions, namely, solutions minimizing Aω,µ among all non-trivial solutions [6, 3]. Alternatively, we can search for solutions to (1.2) with a prescribed L2-norm. Define the energy functional on H2 = H2(RN ,C) by Ep,q(u) := ϵ 2 ∥∆u∥22 + γ 2 ∥∇u∥22 − µ q ∥u∥qq − 1 p ∥u∥pp. It is standard to check that Ep,q is of class C1 and a critical point of Ep,q restricted to the mass constraint S(c) = {u ∈ H2 : ∥u∥22 = c} gives rise to a solution to (1.2) with ∥u∥22 = c. If µ = 0, the corresponding functional is denoted by Ep. When ϵ > 0 and γ > 0, with a pure mass subcritical nonlinearity, i.e., 2 < p < p as considered in [5], the functional Ep has been shown to be bounded from below on S(c), and critical points of E can be sought as global minimizers for any c > 0. Bonheure et al [3] investigated the existence of normalized ground states of (1.2) by exploiting the constrained minimization method and explored the normalized solutions of equation (1.2) with pure mass-critical and mass-supcritical nonlinearity, i.e., p ≤ p < 4∗. When ϵ = 1, γ < 0 and µ = 0, Luo et al [24] used a profile decomposition technique to study the existence of ground states for (1.2) with c = 1 and 2 < p ≤ p. Boussaid et al [9] obtained the existence of normalized ground state solutions for all c > 0, γ < 0 and 2 < p ≤ p without the restriction on c and γ imposed in [24]. For p < p < 4∗, Luo-Yang [25] identified at least two radial normalized solutions: a ground state and an excited state, along with associated asymptotic properties. Recently, Fernández et al [14] utilized the Tomas-Stein inequality to develop a novel approach for establishing non-homogeneous Gagliardo-Nirenberg-type inequalities in RN . These inequalities play a crucial role in proving optimal results regarding the existence of global minimizers for 2 < q ≤ p. Additionally, for the case 2 < q ≤ p, they showed the existence of local minimizers in H2(RN ) but not H2 r (RN ). When ϵ > 0 and γ = 0, equation (1.1) becomes the biharmonic nonlinear Schrödinger equation, in which the stability of solitons in magnetic materials was investigated [16, 37]. Phan [33] presented the existence of normalized ground state solutions of (1.1) for ϵ > 0 and γ = 0 with the pure mass-critical nonlinearity. The case involving mass supercritical nonlinearities was discussed in [27], where normal- ized ground states were shown to exist for 2 < q < p < p = 4∗. The existence of normalized ground state solutions for p ≤ q < p ≤ 4∗ was shown in [28]. EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 3 As for the case ϵ > 0, γ > 0 and µ > 0, however, as far as we know, very little has been known for the mixed dispersion fourth-order nonlinear Schrödinger equation with combined nonlinearities. This constitutes one of our primary motivations of study in the existence of normalized ground state solutions of (1.1) for 2 < q < 2 + 4 N < p < p ≤ 4∗ and p ≤ q < p < 4∗, respectively. For simplicity, we set ϵ = γ = 1. Definition 1.1. We say that a solution uc ∈ S(c) of equation (1.2) is a ground state solution to (1.2) if it possesses the minimal energy among all solutions in S(c), i.e., if Ep,q(uc) = inf{Ep,q(u), u ∈ S(c), (Ep,q|S(c)) ′(u) = 0}. We start with the case 2 < q < 2 + 4 N < p < p ≤ 4∗ by setting V (c) := {u ∈ S(c) : ∥∆u∥22 + |∇u∥22 < ρ0}, ∂V (c) = {u ∈ S(c) : ∥∆u∥22 + |∇u∥22 = ρ0}, where ρ0 is a suitable positive constant. For any given µ > 0, we aim to determine a specific value c0 = c0(µ) > 0 such that for any c ∈ (0, c0) it holds mp,q(c) := inf u∈V (c) Ep,q(u) < 0 < inf u∈∂V (c) Ep,q(u). Theorem 1.2. Let N ≥ 5, µ > 0 and 2 < q < 2+ 4 N < p < p ≤ 4∗. For any µ > 0, there exists c0 = c0(µ) > 0 such that for any c ∈ (0, c0), the constraint functional Ep,q|S(c) admits a ground state, which corresponds to a local minimizer of Ep,q in the set V (c). As p > p, it is evident that the constrained functional Ep,q|S(c) is unbounded from below. However, the presence of the lower order term |u|q−2u with 2 < q < 2 + 4 N creates a geometry of local minima on S(c) for sufficiently small c > 0. The challenge in establishing the existence of local minimizers arises from the lack of compactness of the bounded minimizing sequence {un} ⊂ V (c) due to the noncompact embedding H2(RN ) ↪→ L2(RN ). By employing a minimization approach and incorporating the subadditivity of ground state energy, we overcome this obstacle and demonstrate the existence of local minima. Furthermore, we find that any ground state serves as a local minimum for the associated energy functional. Theorem 1.3. Let N ≥ 5, µ > 0 and p ≤ q < p < 4∗. If q = p, we assume that µc4/N < N+4 NCq N,q . Then there exists a sufficiently small c∗ > 0 such that for any c ∈ (0, c∗), the constrained functional Ep,q|S(c) possesses a critical point u at a positive level Ep,q(u) > 0 with the following properties: u satisfies (1.2) for some ω > 0 and represents a normalized ground state of (1.2) on S(c). We introduce the Nehari-Pohozaev set of Ep,q|S(c) as follows Qp,q(c) = {u ∈ S(c) : Qp,q(u) = 0}, where Qp,q(u) = ∥∆u∥22 + 1 2 ∥∇u∥22 − µγq∥u∥qq − γp∥u∥pp, γr := N(r − 2) 4r = N 2 (1 2 − 1 r ) , ∀r ∈ (2, 4∗]. 4 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 It is easily seen that all critical points of Ep,q|S(c) lie in Qp,q(c). To prove Theorem 1.3, we shall employ a direct minimization method for Ep,q on Qp,q(c). A crucial step is to show the convergence of a minimizing sequence {un} ⊂ Qp,q(c) of Ep,q at mp,q(c). The sign of the Lagrange multiplier ω ∈ R plays a pivotal role in the analysis. However, tackling this issue is challenging because of the presence of the term ∥∇u∥2. As demonstrated in Lemma 4.5, we identify a sufficiently small c∗ > 0 such that for any c ∈ (0, c∗), the corresponding ωc remains positive. Another difficulty comes from weak limits of the minimizing sequence, which may violate the constraint due to the non-compactness of the embedding H2(RN ) ↪→ L2(RN ). Overcoming this obstacle, we need to show that the mapping c 7→ mp,q(c) is strictly decreasing. This, together with the relationship between the energy func- tional Ep,q and the Nehari-Pohozaev functional Qp,q, leads to strong convergence of the minimizing sequence in H2(RN ). Subsequently, by showing that Qp,q(c) is a natural constraint, we observe that the minimizer of Ep,q on Qp,q(c) constitutes a normalized ground state solution of (1.2). The paper is organized as follows. In Section 2, we provide some preliminary concepts and lemmas that will be utilized throughout the paper. We prove Theorem 1.2 in Section 3 and prove Theorem 1.3 in Section 4, respectively. 2. Preliminary results Throughout this article, for 1 ≤ r <∞, Lr(RN ) denotes the standard Lebesgue space with norm ∥u∥rr := ∫ RN |u|rdx. Additionally, the positive constants are denote by C,C1, C2, . . . , with values that may vary from line to line. The open ball in RN is denoted as BR(x) with center at x and radius R. In this section, we present some preliminary results which will be used in the next two sections. We start with recalling the well-known Gagliardo-Nirenberg inequality and Sobolev inequality. Lemma 2.1 ([31]). If N ≥ 5 and 2 < r < 4∗, then the Gagliardo-Nirenberg inequality ∥u∥rr ≤ Cr N,r∥∆u∥ rγr 2 ∥u∥r(1−γr) 2 holds for u ∈ H2(RN ), where CN,r denotes the sharp constant. Lemma 2.2 ([36]). When N ≥ 5, we have S∥u∥24∗ ≤ ∥∆u∥22, ∀u ∈ H2(RN ), where S > 0 depending only on N denotes an optimal constant. Note that the following interpolation inequality holds:∫ RN |∇u|2dx ≤ (∫ RN |∆u|2dx )1/2(∫ RN |u|2dx )1/2 , ∀u ∈ H2(RN ). (2.1) By similar arguments as those in [39], we can obtain the Lions’ type lemma in H2(RN ). Lemma 2.3. Assume that {un} is bounded in H2(RN ). For any R > 0, if sup y∈RN ∫ BR(y) |un|2dx→ 0 as n→ ∞, then un → 0 in Lr(RN ) for r ∈ (2, 4∗). EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 5 To understand the geometry of the constrained functional, we consider the func- tion f(c, ρ) defined on R+ × R+ by f(c, ρ) = 1 2 − µ q Cq N,qρ α0cα1 − Cp N,p p ρα2cα3 , and its restriction gc(ρ) is defined on (0,∞) by ρ 7→ gc(ρ) := f(c, ρ) for each c ∈ (0,∞), where α0 = N(q − 2) 8 − 1, α1 = 2N − q(N − 4) 8 , α2 = N(p− 2) 8 − 1, α3 = 2N − p(N − 4) 8 . Note that for any N ≥ 5 and 2 < q < 2 + 4 N < p < p ≤ 4∗, we have α0 ∈ (−1,− 1 2 ), α1 ∈ (N+4 2N , 1), α2 ∈ (0, 4 N−4 ], and α3 ∈ [0, 4 N ). Lemma 2.4. For each c > 0, the function gc(ρ) has a unique global maximum and the maximum value satisfies max ρ>0 gc(ρ)  > 0 if c < c0, = 0 if c = c0, maxρ>0 gc(ρ) < 0 if c > c0, where c0 = ( 1 2K )N/4 > 0 (2.2) with K = µ q Cq N,q [ − α0 α2 µp q Cq N,q Cp N,p ] α0 α2−α0 + Cp N,p p [ − α0 α2 µp q Cq N,q Cp N,p ] α2 α2−α0 > 0. Proof. From the definition of gc(ρ) it follows that g′c(ρ) = −α0 µ q Cq N,qρ α0−1cα1 − α2 1 p Cp N,pρ α2−1cα3 . Hence, the equation g′c(ρ) = 0 has a unique solution: ρc = [ − α0 α2 µp q Cq N,q Cp N,p ] 1 α2−α0 c α1−α3 α2−α0 . (2.3) Taking into account that gc(ρ) → −∞ as ρ → 0 and gc(ρ) → −∞ as ρ → ∞, we obtain that ρc is the unique global maximum point of gc(ρ) and the maximum value is max ρ>0 gc(ρ) = 1 2 − µ q Cq N,q [ − α0 α2 µp q Cq N,q Cp N,p ] α0 α2−α0 c α0(α1−α3) α2−α0 cα1 − Cp N,p p [ − α0 α2 µp q Cq N,q Cp N,p ] α2 α2−α0 c α2(α1−α3) α2−α0 cα3 = 1 2 − µ q Cq N,q [ − α0 α2 µp q Cq N,q Cp N,p ] α0 α2−α0 c α1α2−α0α3 α2−α0 − Cp N,p p [ − α0 α2 µp q Cq N,q Cp N,p ] α2 α2−α0 c α1α2−α0α3 α2−α0 6 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 = 1 2 −KcN/4. By the definition of c0, we obtain maxρ>0 gc0(ρ) = 0. □ Remark 2.5. When p = 4∗, we use S−4∗/2 instead of Cp N,p, where S is the optimal constant given in Lemma 2.2. Lemma 2.6. Let (c1, ρ1) ∈ R+ × R+ be such that f(c1, ρ1) ≥ 0. Then for any c2 ∈ (0, c1] we have f(c2, ρ2) ≥ 0, if ρ2 ∈ [ c2 c1 ρ1, ρ1]. Proof. Since c→ f(·, ρ) is a non-increasing function, we have f(c2, ρ1) ≥ f(c1, ρ1) ≥ 0. Taking into account α0 + α1 = q−2 2 and α2 + α3 = p−2 2 , we obtain f(c2, c2 c1 ρ1)− f(c1, ρ1) = µ q Cq N,qρ α1 1 cα1 1 (1− ( c2 c1 )α0+α1) + 1 p Cp N,pρ α1 1 cα3 1 (1− ( c2 c1 )α2+α3) = µ q Cq N,qρ α1 1 cα1 1 (1− ( c2 c1 ) q−2 2 ) + 1 p Cp N,pρ α1 1 cα3 1 (1− ( c2 c1 ) p−2 2 ). Since c2 < c1, 2 < q < 2 + 4 N and p < p ≤ 4∗, we derive f(c2, c2 c1 ρ1) ≥ f(c1, ρ1) ≥ 0. We claim that if gc2( c2 c1 ρ) ≥ 0 and gc2(ρ1) ≥ 0, then f(c2, ρ) = gc2(ρ) ≥ 0, for ρ ∈ [ c2 c1 ρ, ρ1]. Indeed, if gc2(ρ) < 0 for some ρ ∈ [ c2c1 ρ, ρ1], then there exists a local minimum point on ( c2c1 ρ, ρ1). This contradicts the fact in Lemma 2.4 that the function gc2(ρ) has a unique critical point which has to be its unique global maximum. □ Lemma 2.7. For p < q < p < 4∗, a > 0, b ≥ 0, c ≥ 0 and d ≥ 0 with c + d > 0, which are independent of t, we denote H(a, b, c, d) = max t>0 { a · t2 + b · t− c · t N(q−2) 4 − d · t N(p−2) 4 } . Then the function (a, b, c, d) 7→ H(a, b, c, d) is continuous. Proof. By making slight modifications to the proof of [2, Lemma 5.2], we can arrive at the desired result. So, we omit the details here. □ 3. Case 2 < q < 2 + 4 N < p < p ≤ 4∗ In this section, we show that ground states of equation (1.2) exist which corre- spond to the local minima of the associated functional. EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 7 3.1. Properties of mapping c 7→ mp,q(c). Let c0 > 0 be determined by equation (2.2) and let ρ0 := ρc0 > 0 be defined by equation (2.3). According to Lemmas 2.4 and 2.6, it follows that f(c0, ρ0) = 0, and f(c, ρ0) > 0 for all c ∈ (0, c0). Set Bρ0 = {u ∈ H2(RN ) : ∥∆u∥22 + ∥∇u∥22 < ρ0} and V (c) := S(c) ∩Bρ0 . For c ∈ (0, c0), we consider the local minimization problem: mp,q(c) = inf u∈V (c) Ep,q(u). Lemma 3.1. Let c ∈ (0, c0) and 2 < q < 2 + 4 N < p < p ≤ 4∗. Then the following three assertions hold. (1) mp,q(c) = infu∈V (c)Ep,q(u) < 0 < infu∈∂V (c)Ep,q(u);. (2) The function c 7→ mp,q(c) is a continuous mapping. (3) For all α ∈ (0, c), we have mp,q(c) ≤ mp,q(α) +mp,q(c− α). If mp,q(α) or mp,q(c− α) is attained, then the inequality is strict. Proof. (1) For any u ∈ ∂V (c), we have ∥∆u∥22 + ∥∇u∥22 = ρ0. Applying the Gagliardo-Nirenberg inequality leads to Ep,q(u) ≥ 1 2 (∥∆u∥22 + ∥∇u∥22)− µ q Cq N,q(∥∆u∥ 2 2 + ∥∇u∥22)α0+1(∥u∥22)α1 − Cp N,p p (∥∆u∥22 + ∥∇u∥22)α2+1(∥u∥22)α3 = (∥∆u∥22 + ∥∇u∥22)f(∥u∥22, ∥∆u∥22 + ∥∇u∥22) = ρ0f(c, ρ0) > ρ0f(c0, ρ0) = 0. (3.1) Let u ∈ S(c) be arbitrary but fixed. For s ∈ R+, set us(x) = sN/2u(sx). Clearly, us ∈ S(c) for any s ∈ R+. We define ψu(s) = Ep,q(us) = s4 2 ∥∆u∥22 + s2 2 ∥∇u∥22 − µ q sN(q−2)/2∥u∥qq − 1 p sN(p−2)/2∥u∥pp, for all s > 0. It is easily seen that ψu(s) → 0− as s → 0. Hence, there exists sufficiently small s0 > 0 such that ∥∆us0∥22 + ∥∇us0∥22 < ρ0 and Ep,q(us0) = ψu(s0) < 0. Consequently, we have mp,q(c) < 0. (2) Let c ∈ (0, c0) be arbitrary and {cn} ⊂ (0, c0) be such that cn → c. By the definition of mp,q(cn) with mp,q(cn) < 0, for any ϵ > 0 small enough, there exists un ∈ V (c) such that Ep,q(un) ≤ mp,q(cn) + ϵ and Ep,q(un) < 0. (3.2) Let zn = √ c cn un. Clearly, zn ∈ S(c). On the one hand, if cn ≥ c, then ∥∆zn∥22 + ∥∇zn∥22 = c cn (∥∆un∥22 + ∥∇un∥22) < ρ0. On the other hand, if cn < c, by Lemma 2.6 and f(cn, ρ0) ≥ f(c0, ρ0) = 0, we have f(cn, ρ) ≥ 0 for any ρ ∈ [ cnc ρ0, ρ0]. However, from (3.1) and (3.2) it follows that f(∥un∥22, ∥∆un∥22 + ∥∇un∥22) < 0. Hence, ∥∆un∥22 + ∥∇un∥22 < cn c ρ0 and ∥∆zn∥22 + ∥∇zn∥22 < c cn · cn c ρ0 = ρ0. Since zn ∈ V (c), we have mp,q(c) ≤ Ep,q(zn) 8 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 = Ep,q(un) + (Ep,q(zn)− Ep,q(un)) = Ep,q(un) + 1 2 ( c cn − 1)∥∆un∥22 + 1 2 ( c cn − 1)∥∇un∥22 − µ q [( c cn ) q 2 − 1]∥un∥qq − 1 p [( c cn )p/2 − 1]∥un∥pp. That is, mp,q(c) ≤ Ep,q(zn) = Ep,q(un) + on(1) as n→ ∞. (3.3) Using (3.2) and (3.3) yields mp,q(c) ≤ mp,q(cn) + ϵ+ on(1). Now, we let u ∈ V (c) be such that Ep,q(u) ≤ mp,q(c) + ϵ and Ep,q(u) < 0. Set un := √ cn c u. Then un ∈ S(cn), and cn → c implies that ∥∆un∥22+∥∇un∥22 < ρ0 for n large enough. So un ∈ V (cn). Note that Ep,q(un) → Ep,q(u). Thus, we obtain mp,q(cn) ≤ Ep,q(u) + (Ep,q(un)− Ep,q(u)) ≤ mp,q(c) + ϵ+ on(1). Because of the arbitrariness of ϵ > 0, we infer that mp,q(cn) → mp,q(c). (3) Given α ∈ (0, c), it suffices to prove that ∀θ ∈ (1, c α ] : mp,q(θα) ≤ θmp,q(α) and that, if mp,q(α) is attained, the inequality is strict. Using (i), for any ϵ > 0 small enough, there exists u ∈ V (α) such that Ep,q(u) ≤ mp,q(α) + ϵ and Ep,q(u) < 0. From Lemma 2.6 and f(α, ρ0) ≥ f(c0, ρ0) = 0, it follows that f(α, ρ) ≥ 0 for any ρ ∈ [αc ρ0, ρ0]. Hence, using (3.1) and (3.2) we obtain f(∥u∥22, ∥∆u∥22 + ∥∇u∥22) < 0. That is, ∥∆u∥22 + ∥∇u∥22 < α c ρ0. Set v = √ θu. Then ∥v∥22 = θα and ∥∆v∥22 + ∥∇v∥22 < ρ0. Thus v ∈ V (θα). A direct calculation yields mp,q(θα) ≤ Ep,q(v) < 1 2 θ∥∆u∥22 + 1 2 θ∥∇u∥22 − µ q θ∥v∥qq − 1 p θ∥v∥pp = θEp,q(u) ≤ θ(mp,q(α) + ϵ). Because of the arbitrariness of ϵ, we obtain mp,q(θα) ≤ θmp,q(α). If mp,q(α) is attained, we can choose ϵ = 0. □ 3.2. Proof of Theorem 1.2. We define Mc = {u ∈ V (c) : Ep,q(u) = mp,q(c)}. Lemma 3.2. Let 2 < q < 2 + 4 N < p < p ≤ 4∗. For any c ∈ (0, c0) and the sequence {un} ⊂ Bρ0 such that ∥un∥2 → c and Ep,q(un) → mp,q(c), there exists a sequence {yn} ⊂ RN such that for some R > 0 it holds∫ BR(yn) |un|2dx ≥ β > 0. (3.4) EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 9 Proof. By way of contradiction, we assume that (3.4) does not hold. From {un} ⊂ Bρ0 and ∥un∥2 → c it follows that {un} is bounded in H2(RN ). For 2 < q < 2 + 4 N < p < p < 4∗, by Lemma 2.3, we deduce that ∥un∥qq → 0 and ∥un∥pp → 0, as n → ∞. At this point, it follows that Ep,q(un) ≥ on(1). If p = 4∗, in view of f(c0, ρ0) = 0, a straightforward computation yields Ep,q(un) = 1 2 ∥∆un∥22 + 1 2 ∥∇un∥22 − 1 4∗ ∥un∥4 ∗ 4∗ + on(1) ≥ 1 2 ∥∆un∥22 + 1 2 ∥∇un∥22 − 1 4∗ 1 S4∗/2 (∥∆un∥22 + ∥∇un∥22) 4∗ 2 + on(1) ≥ (∥∆n∥22 + ∥∇un∥22)( 1 2 − 1 4∗ 1 S4∗/2 ρα2 0 ) + on(1) = (∥∆n∥22 + ∥∇un∥22) µ q Cq N,qρ α0 0 cα1 0 + on(1) > 0. Both cases contradict the factmp,q(c) < 0. Thus, we arrive at the desired result. □ Proposition 3.3. For any c ∈ (0, c0), if {un} ⊂ Bρ0 is such that ∥un∥22 → c and Ep,q(un) → mp,q(c), then, up to translation, un −→ uc ∈ Mc in H2(RN ). In particular, the set Mc is compact in H2(RN ), up to translation. The proof of the above proposition can be obtained by similar arguments as in [27] (see also [18]). Proposition 3.4. For any c ∈ (0, c0), if mp,q(c) is reached, then any ground state is contained in V (c). Proof. For any v ∈ S(c) and s ∈ (0,∞), we obtain ψ′ v(s) = 2 s Q(vs), which implies that if w ∈ S(c) is a ground state solution, then there exist v ∈ S(c) and s0 > 0 such that w = vs0 , Ep,q(w) = ψv(s0) and ψ ′ v(s0) = 0. To conclude the proof, it suffices to show that ψ′ v(s) has at most two zeros. This is equivalent to showing that the function s 7→ ψ′ v(s) s has at most two zeros. Note that ξ(s) = ψ′ v(s) s = 2s2∥∆u∥22 + ∥∇u∥22 − s N(q−2) 2 −2µN(q − 2) 2q ∥u∥qq − s N(p−2) 2 −2N(p− 2) 2p ∥u∥pp and ξ′(s) = s[4∥∆u∥22 − s N(q−2) 2 −4 · µN(q − 2) 2q ( N(q − 2) 2 − 2)∥u∥qq − s N(p−2) 2 −4 · N(p− 2) 2p ( N(p− 2) 2 − 2)∥u∥pp] =: s[4∥∆u∥22 − f(s)]. So we need to show that ξ′(s) is the unique solution. Since 2 < q < 2 + 4 N < p < p ≤ 4∗, N ≥ 5 and s > 0, it is easy to see that s → f(s) is a non-increasing function. Hence, ξ′(s) has a unique solution and ξ(s) has at most two zeros. 10 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 Now, since ψv(s) → 0−, ∥∆vs∥22 + ∥∇vs∥22 → 0 as s→ 0 and ψv(s) = Ep,q(vs) > 0, when vs ∈ ∂V (c), ψ′ v has a first zero s1 > 0 corresponding to a local minima. Also, from ψv(s1) < 0, ψv(s) > 0 when vs ∈ ∂V (c) and ψv(s) → −∞ as s→ ∞, ψv has a second zero s2 > s1 corresponding to a local maxima. In particular, vs1 ∈ V (c) and Ep,q(vs1) = ψv(s1) < 0. Thus, if mp,q(c) is achieved, it is a ground state level. □ Proof of Theorem 1.2. The existence of a minimizer for Ep,q on V (c) follows from Proposition 3.3. By Proposition 3.4, this local minimizer is a ground state. □ 4. Case p ≤ q < p < 4∗ In this section, we present the proof of Theorem 1.3. 4.1. Monotonicity of ground state energy mp,q(c). We start by showing some properties of Qp,q(c) and the energy functional Ep,q restricted on it. For any u ∈ S(c) and s ∈ (0,+∞), we define us(x) = sN/4u( √ sx), for a.e. x ∈ RN . Clearly, us ∈ S(c) for any s > 0. It follows that Ep,q(us) = s2 2 ∥∆u∥22 + s 2 ∥∇u∥22 − µ q s N(q−2) 4 ∥u∥qq − 1 p s N(p−2) 4 ∥u∥pp and Qp,q(us) = s2∥∆u∥22 + s 2 ∥∇u∥22 − µγqs N(q−2) 4 ∥u∥qq − γps N(p−2) 4 ∥u∥pp. Then, we have the following properties for Ep,q(us) and Qp,q(us). Lemma 4.1. . Let N ≥ 5, c > 0, µ > 0 and p ≤ q < p < 4∗. When q = p, we assume that µc4/N < N+4 NCq N,q . Then for any u ∈ S(c), there exists a unique su ∈ (0,+∞) such that usu ∈ Qp,q(c) and su is the unique critical point of Ep,q(us) such that Ep,q(usu) = maxs∈(0,+∞)Ep,q(us). The function u 7→ Ep,q(usu) is concave on [su,+∞). In particular, if Qp,q(u) ≤ 0, then su ∈ (0, 1]. Moreover, the map u 7→ su is of class C1. Since the proof is similar to the one of [28, Lemma 3.4], we omit it here. Under the same assumptions described in Lemma 4.1, we can obtain the following results concerning the Nehari-Pohozaev’s type set Qp,q(c) and the constrained functional Ep,q. Lemma 4.2. Let N ≥ 5, c > 0, µ > 0 and p ≤ q < p < 4∗. When q = p, we assume that µc4/N < N+4 NCq N,q . Then we have (1) Qp,q(c) ̸= ∅; (2) infu∈Qp,q(c) ∥∆u∥22 + 1 2∥∇u∥ 2 2 > 0 and infu∈Qp,q(c) ∥∆u∥22 > 0; (3) infu∈Qp,q(c)Ep,q(u) > 0; (4) Ep,q is coercive on Qp,q(c). Proof. (1) By Lemma 4.1, for any u ∈ S(c), there always exists su > 0 such that usu ∈ Qp,q(c), it follows that Qp,q(c) ̸= ∅. (2) For any u ∈ Qp,q(c), using the Gagliardo-Nirenberg inequality yields ∥∆u∥22 + 1 2 ∥∇u∥22 = µγq∥u∥qq + γp∥u∥pp EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 11 ≤ µγqC q N,q( √ c)q(1−γq)(∥∆u∥2 + 1 2 ∥∇u∥22) qγq 2 + γpC p N,p( √ c)p(1−γp)(∥∆u∥2 + 1 2 ∥∇u∥22) pγp 2 . If p < q < p, then pγp > qγq > 2. If p = q < p and µc4/N < N+4 NCq N,q , then pγp > qγq = 2 and µNCq N,q N+4 c4/N < 1. In either case, there exists a constant C > 0 such that ∥∆u∥22 + 1 2∥∇u∥ 2 2 ≥ C, which implies infu∈Qp,q(c) ∥∆u∥22 + 1 2∥∇u∥ 2 2 > 0. By a similar argument, we can deduce that infu∈Qp,q(c) ∥∆u∥22 > 0. (3) For eachy u ∈ Qp,q(c), we have Ep,q(u) = qγq − 2 2qγq ∥∆u∥22 + qγq − 1 2qγq ∥∇u∥22 + pγp − qγq pqγq ∥u∥pp. (4.1) From (2) it follows that infu∈Qp,q(c)Ep,q(u) > 0. (4) By (4.1), it is easily seen that (4) holds. □ For any fixed c > 0, Lemma 4.2 indicates that mp,q(c) = inf u∈Qp,q(c) Ep,q(u) is well-defined and strictly positive. We now analyze the behaviors of mp,q(c) when c > 0 varies. Lemma 4.3. Let p ≤ p < q < 4∗. When q = p, we assume that µc4/N < N+4 NCq N,q . Then the function c 7→ mp,q(c) is continuous for c ∈ (0,+∞). Proof. We define γ(c) = inf u∈S(c) max s>0 Ep,q(us). (4.2) To prove γ(c) = mp,q(c), for any u ∈ Qp,q(c) we have Ep,q(u) = maxs>0Ep,q(us), which implies that γ(c) ≤ mp,q(c). On the other hand, for any u ∈ S(c), by Lemma 4.1 there exists su > 0 such that usu ∈ Qp,q(c) and maxs>0Ep,q(us) = Ep,q(usu) ≥ mp,q(c). Thus, we have γ(c) = mp,q(c). For each fixed c > 0, taking {cn} ⊂ R+ such that cn → c, we shall prove limn→∞mp,q(cn) = mp,q(c). For any ϵ > 0, by the definition of mp,q(c) there exists v ∈ Qp,q(c) such that Ep,q(v) ≤ mp,q(c) + ϵ 2 . Set vn := √ cn c v ∈ S(cn). From the fact µc4/N < N+4 NCp N,p , cn → c and Lemma 2.7, it follows that mp,q(cn) ≤ max s>0 Ep,q((vn)s) = max s>0 ( s2 2 ∥∆vn∥22 + s 2 ∥∇vn∥22 − µ q s N(q−2) 4 ∥vn∥qq − 1 p s N(p−2) 4 ∥vn∥pp) ≤ max s>0 ( s2 2 ∥∆v∥22 + s 2 ∥∇v∥22 − µ q s N(q−2) 4 ∥v∥qq − 1 p s N(p−2) 4 ∥v∥pp) + ϵ 2 = max s>0 Ep,q((v)s) + ϵ 2 = Ep,q(v) + ϵ 2 ≤ mp,q(c) + ϵ. That is, lim sup n→∞ mp,q(cn) ≤ mp,q(c). (4.3) 12 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 Then we take {un} ⊂ Qp,q(cn) such that Ep,q(un) ≤ mp,q(cn) + ϵ 2 . (4.4) In view of Qp,q(un) = 0, for n large enough, from (4.3) and (4.4) it follows that(1 2 − 1 qγq ) ∥∆un∥22 + 1 2 ( 1− 1 qγq ) ∥∇un∥22 + ( γp qγq − 1 p ) ∥un∥pp ≤ Ep,q(un) ≤ mp,q(cn) + ϵ 2 ≤ mp,q(c) + 3ϵ 4 . If pγp > qγq > 2, we can derive that {un} is bounded in H2(RN ). If pγp > qγq = 2, recalling Lemma 4.2 (2), we can see the same result. Without loss of generality, as n→ ∞ we assume that ∥∆un∥22 → C1, ∥∇un∥22 → C2, ∥un∥qq → C3, ∥un∥pp → C4. If follows from Lemma 4.2 (2) that C1 > 0, C2 ≥ 0, and C3 ≥ 0, C4 ≥ 0 with C3 + C4 > 0. Let ũn := √ c cn un. Clearly, ũn ∈ S(c). From Lemma 2.7 it follows that mp,q(c) ≤ max s>0 Ep,q((ũn)s) = max s>0 [s2 2 ( c cn )∥∆un∥22 + s 2 ( c cn )∥∇un∥22 − µ q s N(q−2) 4 ( c cn ) q 2 ∥un∥qq − 1 p s N(p−2) 4 ( c cn )p/2∥un∥pp ] ≤ max s>0 ( s2 2 ∥∆un∥22 + s 2 ∥∇un∥22 − µ q s N(q−2) 4 ∥un∥qq − 1 p s N(p−2) 4 ∥un∥pp) + 3ϵ 4 = max s>0 Ep,q((un)s) + 3ϵ 4 = Ep,q(un) + 3ϵ 4 ≤ mp,q(cn) + ϵ. That is, mp,q(c) ≤ lim inf n→∞ mp,q(cn). Hence, we arrive at the desired result. □ Lemma 4.4. Let p ≤ p < q < 4∗. When q = p, we assume that µc4/N < N+4 NCq N,q . Then the function c 7→ mp,q(c) is non-increasing for c ∈ (0,+∞). Proof. For 0 < c1 < c2 < +∞, we shall prove that mp,q(c2) ≤ mp,q(c1). According to the definition of γ(c) in 4.2, for any ϵ > 0 there exists u1 ∈ Qp,q(c1) such that Ep,q(u1) ≤ mp,q(c1) + ϵ 2 and max λ>0 Ep,q((u1)λ) = E(u1). For κ > 0 and λ ∈ (0, 1), we define wκ λ := uκ1 + (vκ0 )λ. We choose uκ1 ∈ H2(RN ) such that suppuκ1 ⊂ B 1 κ (0) and ∥uκ1 − u1∥ = o(κ), while vδ0 := (c2 − ∥uκ1∥22)1/2 vκ ∥vκ∥2 , where vκ ∈ C∞ 0 (RN ) such that supp vκ ⊂ EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 13 B 2 κ+1(0)\B 2 κ (0). It is obvious that dist(supp(vκ0 )λ, suppu κ 1 ) ≥ 1 κ ( 2√ λ − 1) > 0. Hence, ∥wκ λ∥22 = c2. By a standard argument, as λ, κ→ 0, we derive ∥∆wκ λ∥22 → ∥∆u1∥22, ∥∇wκ λ∥22 → ∥∇u1∥22, ∥wκ λ∥qq → ∥u1∥qq, ∥wκ λ∥pp → ∥u1||pp. Letting (wκ λ)t = tN/4wκ λ( √ tx), by Lemma 2.7 again, we can deduce that for λ, κ > 0 small enough, it holds mp,q(c2) ≤ max t>0 Ep,q((w κ λ)t) ≤ max t>0 Ep,q((u1)t) + ϵ 2 = Ep,q(u1) + ϵ 2 ≤ mp,q(c1) + ϵ. □ Lemma 4.5. Let p ≤ q < p < 4∗. Assume that uc ∈ S(c) solves ∆2u−∆u+ ωcu = µ|u|q−2u+ |u|p−2u. (4.5) Then there exists c∗ > 0 such that ωc > 0 for any c ∈ (0, c∗). Proof. By (4.5) we deduce Qp,q(u) = 0 and ∥∆uc∥22 + ∥∇uc∥22 + ωc∥uc∥22 − µ∥uc∥qq − ∥uc∥pp = 0. Then ωcγqc = (1− γq)∥∆uc∥22 + ( 1 2 − γq)∥∇uc∥22 − (γp − γq)∥uc∥pp. (4.6) For small c > 0, using the Gagliardo-Nirenberg inequality leads to ∥∆uc∥22 = γpC q N,q∥∆uc∥ qγq 2 ( √ c)q(1−γq) + γpC p N,p∥∆uc∥ pγp 2 ( √ c)p(1−γp) ≤ γp max{Cq N,q, C p N,p}( √ c)q(1−γq)(∥∆uc∥ qγq 2 + ∥∆uc∥ pγp 2 ). Then, for p ≤ q < p < 4∗, as c→ 0 we obtain∫ RN |∆uc|2dx→ ∞. (4.7) On the other hand, we from (2.1) and (4.6) derive ωcγqc = (1− γq)∥∆uc∥22 + ( 1 2 − γq)∥∇uc∥22 − (γp − γq)∥uc∥pp > (1− γq)∥∆uc∥22 + ( 1 2 − γq) √ c∥∆uc∥2. From (4.7), it follows that ωc > 0 if c > 0 is small enough. □ Lemma 4.6. Let p ≤ q < p < 4∗ and c ∈ (0, c∗). When q = p, we assume that µc4/N < N+4 NCq N,q . Suppose that u ∈ S(c) such that Ep,q(u) = mp,q(c) and ∆2u−∆u+ ωu = µ|u|q−2u+ |u|p−2u. Then the function c 7→ mp,q(c) is strictly decreasing in a right neighborhood of c. Proof. By Lemma 4.5, we know that ω > 0. Set uλ,t(x) = tN/4 √ λu( √ tx) for λ, t > 0. We define K(λ, t) = Ep,q(uλ,t) = t2 2 λ∥∆u∥22+ t 2 λ∥∇u∥22− µ · t N(q−2) 4 q λ q 2 ∥u∥qq− t N(p−2) 4 p λp/2∥u∥pp and M(λ, t) = Qp,q(uλ,t) = t2λ∥∆u∥22 + t 2 λ∥∇u∥22 − µγqt N(q−2) 4 λ q 2 ∥u∥qq − γpt N(p−2) 4 λp/2∥u∥pp. 14 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 By a direct calculation, we have ∂K ∂λ (1, 1) = 1 2 ∥∆u∥22 + 1 2 ∥∇u∥22 − µ 2 ∥u∥qq − 1 2 ∥u∥pp = −1 2 ωc, ∂K ∂t (1, 1) = ∥∆u∥22 + 1 2 ∥∇u∥22 − µγq∥u∥qq − γp∥u∥pp = 0, ∂2K ∂t2 (1, 1) = ∥∆u∥22 − µγq( N(q − 2) 4 − 1)∥u∥qq − γp( N(p− 2) 4 − 1)∥u∥pp < 0, which yields for δt small enough and δλ > 0, K(1 + δλ, 1 + δt) < K(1, 1) for ω > 0. (4.8) In addition, we observe that M(1, 1) = Qp,q(u) = ∥∆u∥22 + 1 2 ∥∇u∥22 − µγq∥u∥qq − γp∥u∥pp = 0. We now claim that ∂M ∂t (1, 1) = 2∥∆u∥22 + 1 2 ∥∇u∥22 − µγq N(q − 2) 4 ∥u∥qq − γp N(p− 2) 4 ∥u∥pp ̸= 0. Otherwise, we assume that ∂M ∂t (1, 1) = ∥∆u∥22 + 1 4 ∥∇u∥22 − µγq N(q − 2) 8 ∥u∥qq − γp N(p− 2) 8 ∥u∥pp = 0. Then for any p ≤ q < p < 4∗, we have that 1 4 ∥∇u∥22 = µγq(1− N(q − 2) 8 )∥u∥qq + γp(1− N(p− 2) 8 )∥u∥pp, which is impossible. According to the implicit function theorem, we deduce that there exists ϵ > 0 and a continuous function g : [1−ϵ, 1+ϵ] 7→ R satisfying g(1) = 1 such that M(λ, g(λ)) = 0 for λ ∈ [1− ϵ, 1 + ϵ]. This together with (4.8) gives mp,q((1 + ϵ)c) ≤ Ep,q(u1+ϵ,g(1+ϵ)) < Ep,q(u) = mp,q(c). We have arrived at the desired result. □ 4.2. Ground states. In this subsection, before presenting the proof of Theorem 1.3, we show the minimizer of Ep,q(u) constrained on Qp,q(c). For convenience, we set f(s) = µ|s|q−2s+ |s|p−2s, F (s) = µ q |s| q + 1 p |s| p and H(s) = f(s)s− 2F (s). Lemma 4.7. Let p ≤ q < p < 4∗ and c ∈ (0, c∗). When q = p, we assume that µc4/N < N+4 NCq N,q . Then there exists u0 ∈ Qp,q(c) such that Ep,q(u0) = mp,q(c). Proof. Using the Ekeland variational principle, there exists a minimizing sequence {un} ⊂ Qp,q(c) such that Ep,q(un) → mp,q(c) as n→ +∞. (4.9) By Lemma 4.2(4), it follows that {un} is bounded in H2(RN ). We claim that {un} is non-vanishing. Indeed, if {un} is vanishing, then it follows from Lemma 2.1 that∫ RN |un|rdx→ 0, for r ∈ (2, 4∗). Since Qp,q(un) = 0 and p ≤ q < p < 4∗, it follows that |∆un|2 + 1 2 |∇un|2 = µγq∥un∥qq + γp∥un∥pp → 0, as n→ ∞, EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 15 which contradicts Lemma 4.2(2). Thus, up to a subsequence, we obtain that un ⇀ u0 ̸= 0 in H2(RN ). Denote un,0 = un − u0. It is easily seen that ∥un∥22 = ∥u0∥22 + ∥un,0∥22 + on(1), ∥∇un∥22 = ∥∇u0∥22 + ∥∇un,0∥22 + on(1), ∥∆un∥22 = ∥∆u0∥22 + ∥∆un,0∥22 + on(1). By the splitting properties of Brezis-Lieb we have H(un) = H(u0) +H(un,0) + on(1), (4.10) Ep,q(un) = Ep,q(u0) + Ep,q(un,0) + on(1), (4.11) Qp,q(un) = Qp,q(u0) +Qp,q(un,0) + on(1). (4.12) We claim that Qp,q(u0) ≤ 0. Up to a subsequence, we assume that δn :=∫ RN |∆un,0|2dx+ 1 2 ∫ RN |∇un,0|2dx→ δ0 ≥ 0. Now we need to consider two cases. Case 1. δ0 = 0. By Lemma 2.3, for any r ∈ (2, 4∗), we have ∫ RN |un,0|rdx → 0. Then Qp,q(un,0) → 0 as n→ +∞. Hence, from (4.12) we derive Qp,q(u0) = 0. Case 2. δ0 > 0. By contradiction, we suppose that Qp,q(u0) > 0. From (4.12) it follows that Qp,q(un,0) ≤ 0. According to Lemma 4.1, there exists sun,0 ∈ (0, 1] such that Qp,q((un,0)sun,0 ) = 0. In view of the fact that H(s) |s|2+ 8 N is strictly increasing for s ∈ (0,∞), we deduce Ep,q(un,0)− Ep,q((un,0)sun,0 ) = 1− s2un,0 2 ∫ RN |∆un,0|2dx+ 1− sun,0 2 ∫ RN |∇un,0|2dx − ∫ RN F (un,0)dx+ s−N/2 un,0 ∫ RN F (sN/4 un,0 un,0)dx = 1− s2un,0 2 Qp,q(un,0) + ( 1− sn,0 2 − 1− s2n,0 4 ) ∫ RN |∇un,0|2dx + 1− s2n,0 2 N 4 ∫ RN (f(un,0)un,0 − 2F (un,0))dx − ∫ RN F (un,0)dx+ s−N/2 un,0 ∫ RN F (sN/4 un,0 un,0)dx ≥ 1− s2n,0 2 N 4 ∫ RN (f(un,0)un,0 − 2F (un,0))dx − ∫ RN F (un,0)dx+ s−N/2 un,0 ∫ RN F (sN/4 un,0 un,0)dx+ 1− s2un,0 2 Qp,q(un,0) = ∫ RN ∫ 1 sn,0 N 4 s|un,0|2+ 8 N ( H(un,0) |un,0|2+ 8 N − H(sN/4un,0) |sN/4un,0|2+ 8 N ) dsdx + 1− s2un,0 2 Qp,q(un,0) ≥ 1− sun,0 2 Qp,q(un,0). 16 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 We denote cn,0 := ∥un,0∥22. Clearly, cn,0 ≤ c. From Lemma 4.4 we derive mp,q(c) = lim n→+∞ ( Ep,q(un)− 1 2 Qp,q(un) ) = lim n→+∞ [(N 8 ∫ RN H(un)dx− ∫ RN F (un)dx ) + 1 4 ∥∇un∥22 ] = (N 8 ∫ RN H(u0)dx− ∫ RN F (u0)dx+ 1 4 ∥∇u0∥22 ) + lim n→+∞ (N 8 ∫ RN H(un,0)dx− ∫ RN F (un,0)dx+ 1 4 ∥∇un,0∥22 ) = [N 8 ∫ RN ( f(u0)u0 − ( 2 + 8 N ) F (u0) ) dx+ 1 4 ∥∇u0∥22 ] + lim n→+∞ ( Ep,q(un,0)− 1 2 Qp,q(un,0) ) ≥ lim n→+∞ ( Ep,q(un,0)− 1 2 Qp,q(un,0) ) ≥ lim n→+∞ ( Ep,q((un,0)sun,0 )− s2un,0 2 Qp,q(un,0) ) ≥ lim n→+∞ Ep,q((un,0)sun,0 ) ≥ lim n→+∞ mp,q(cn,0) ≥ mp,q(c). This indicates that limn→+∞Qp,q(un,0) = 0 and lim n→+∞ Ep,q(un,0) = lim n→+∞ mp,q(cn,0) = mp,q(c). (4.13) On the other hand, combining (4.9) and (4.11) yields mp,q(c) = Ep,q(un) + on(1) = Ep,q(u0) + Ep,q(un,0) + on(1). In view of Ep,q(u0) > 0, from (4.13) it follows that mp,q(c) > mp,q(c)− Ep,q(u0) = lim n→+∞ Ep,q(un,0) = lim n→+∞ mp,q(cn,0) = mp,q(c). This yields a contradiction. Using Qp,q(u0) ≤ 0 and similar arguments as above, there exists s0 ∈ (0, 1] such that (u0)s0 ∈ Qp,q(c0) and Ep,q(u0)− Ep,q((u0)s0) ≥ 1− s20 2 Qp,q(u0). (4.14) We denote c0 = ∥u0∥22. Clearly, c0 ∈ (0, c]. By (4.14) and Lemma 4.4 we have mp,q(c) = lim n→+∞ ( Ep,q(un)− 1 2 Qp,q(un) ) = lim n→+∞ [(N 8 ∫ RN H(un)dx− ∫ RN F (un)dx ) + 1 4 ∥∇un∥22 ] = lim n→+∞ [N 8 ∫ RN ( f(un,0)un,0 − ( 2 + 8 N ) F (un,0) ) dx + 1 4 ∥∇un,0∥22 ] + ( Ep,q(u0)− 1 2 Qp,q(u0) ) ≥ Ep,q((u0)s0)− s20 2 Qp,q(u0) EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 17 ≥ mp,q(c0) ≥ mp,q(c), which implies mp,q(c0) = mp,q(c) and Qp,q(u0) = 0, that is, s0 = 1. Thus we have u0 ∈ Qp,q(c0) and Ep,q(u0) = mp,q(c0). Using Lemma 4.6 at c0 and mp,q(c0) = mp,q(c), we obtain c0 = c and thus Ep,q(u0) = mp,q(c). □ Proof of Theorem 1.3. Consider the functional Ψ(u) : S(c) → R defined by Ψ(u) := Ep,q(usu) = 1 2 s2u∥∆u∥22 + su 2 ∥∇u∥22 − µ q s N(q−2) 4 u ∥u∥qq − 1 p s N(p−2) 4 u ∥u∥pp, where su is given in Lemma 4.1 and usu ∈ Qp,q(c). According to Lemma 4.7, we find u0 ∈ Qp,q(c) such that Ep,q(u0) = mp,q(c). Then there exists v0 ∈ S(c) such that (v0)sv0 = u0 and Ψ(v0) = Ep,q((v0)sv0 ) = Ep,q(u0) = mp,q(c). This implies that v0 is a minimizer of Ep,q restricted on S(c). We claim that Ψ is of class C1 and dΨ(u)[φ] = dEp,q(usu)[φsu ] (4.15) for any u ∈ S(c) and φ ∈ TuS(c). In fact, by the definition of Ψ we have Ψ(u+ tφ)−Ψ(u) = Ep,q((u+ tφ)st)− Ep,q(us0), where |t| is small enough, st = su+tφ and s0 = su is the unique maximum point of the functional Ep,q(us). By the mean value theorem we obtain Ep,q((u+ tφ)st)− Ep,q(us0) ≤ Ep,q((u+ tφ)st)− Ep,q(ust) = s2t 2 ( ∫ RN 2t∆u ·∆φ+ t2|∆φ|2dx) + st 2 ( ∫ RN 2t∇u · ∇φ+ t2|∇φ|2dx) − µs N(q−2) 4 t ∫ RN (∫ 1 0 |u+ sηtφ|q−2(u+ tηtφ)tφdt ) dx − s N(p−2) 4 t ∫ RN (∫ 1 0 |u+ tηtφ|p−2(u+ tηtφ)tφdt ) dx, (4.16) where ηt ∈ (0, 1). Similarly, we derive Ep,q((u+ tφ)st)− Ep,q(us0) ≥ Ep,q((u+ tφ)s0)− Ep,q(us0) = s20 2 ( ∫ RN 2t∆u ·∆φ+ t2|∆φ|2dx) + s0 2 ( ∫ RN 2t∇u · ∇φ+ t2|∇φ|2dx) − µs N(q−2) 4 0 ∫ RN (∫ 1 0 |u+ tθtφ|q−2(u+ tθtφ)tφdt ) dx − s N(p−2) 4 0 ∫ RN (∫ 1 0 |u+ tθtφ|p−2(u+ tθtφ)tφdt ) dx, (4.17) where θt ∈ (0, 1). Since the map u 7→ su is of class C1, from (4.16) and (4.17) it follows that lim t→0 Ψ(u+ tφ)−Ψ(u) t = s2u ∫ RN ∆u ·∆φdx+ su ∫ RN ∇u · ∇φdx 18 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 − s N(q−2) 4 u µ ∫ RN |u|q−2u · φdx− s N(p−2) 4 u µ ∫ RN |u|p−2u · φdx. So the Gâteaux derivative of Ψ is bounded linear in φ and continuous in u. Therefore, Ψ is of class C1. In particular, by changing variables in the integrals, we have dΨ(u)[φ] = s2u ∫ RN ∆u ·∆φdx+ su ∫ RN ∇u · ∇φdx − s N(q−2) 4 u µ ∫ RN |u|q−2u · φdx− s N(p−2) 4 u ∫ RN |u|p−2u · φdx = ∫ RN ∆usu ·∆φsudx+ ∫ RN ∇usu · ∇φsudx − µ ∫ RN |usu |q−2usu · φsudx− ∫ RN |usu |p−2usu · φsudx. = dEp,q(usu)[φsu ]. So the claim (4.15) is true, from which we deduce ∥dEp,q(u0)∥(Tu0S(c))∗ = sup φ∈Tu0 S(c),∥φ∥≤1 |dEp,q(u0)[φ]| = sup φ∈Tu0 S(c),∥φ∥≤1 |dEp,q((v0)sv0 )[(φs−1 v0 )sv0 ]| = sup φ∈Tu0 S(c),∥φ∥≤1 |dΨ(v0)[φs−1 v0 ]| ≤ ∥dΨ(v0)∥(Tv0S(c))∗ · sup φ∈Tu0 S(c),∥φ∥≤1 ∥φs−1 v0 ∥ ≤ max{s−1 v0 , 1}∥dEp,q(v0)∥(Tv0S(c))∗ = 0. It follows that u0 is a critical point of Ep,q restricted on S(c). By Lemma 4.5 for some ω > 0, u0 weakly solves (1.2). In view of Ep,q(u0) = mp,q(c), we infer that u0 is a normalized ground state solution of problem (1.2). □ Acknowledgments. This work is supported by National Natural Science Foun- dation of China No. 11971095. References [1] T. Bartsch, N. Soave; A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037. [2] J. Bellazzini, L. Jeanjean, T. Luo; Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc. (3), 107 (2013), 303-339. [3] D. Bonheure, J.-B. Casteras, T. Gou, L. Jeanjean; Normalized solutions to the mixed disper- sion nonlinear Schrödinger equation in the mass critical and subcritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212. [4] D. Bonheure, J.-B. Casteras, T. Gou, L. Jeanjean; Strong instability of ground states to a fourth order Schrödinger equation, Int. Math. Res. Not. IMRN, (2019), 5299-5315. [5] D. Bonheure, J.-B. Casteras, E. Moreira dos Santos, R. Nascimento; Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation, SIAM J. Math. Anal., 50 (2018), 5027-5071. [6] D. Bonheure, R. Nascimento; Waveguide solutions for a nonlinear Schrödinger equations with mixed dispersion, in: Contributions to nonlinear elliptic equations and systems, Progr. Nonlinear Differential Equations Appl., 86, Birkhäuser/Springer, Cham, (2015), 31-53. EJDE-2024/29 DISPERSION NONLINEAR SCHRÖDINGER EQUATION 19 [7] J. Borthwick, X. Chang, L. Jeanjean, N. Soave; Normalized solutions of L2-supercritical NLS equations on noncompact metric graphs with localized nonlinearities, Nonlinearity, 36, (2023), 3776-3795. [8] T. Boulenger, E. Lenzman; Blowup for biharmonic NLS, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 503-544. [9] N. Boussäıd, A. J. Fernández, L. Jeanjean; Some remarks on a minimization problem associ- ated to a fourth order nonlinear Scrhödinger equation, arXiv.1910.13177. [10] T. Cazenave, P.-L. Lions; Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [11] L. Cely; Stability of ground states of nonlinear Schrodinger systems, Electron. J. Differential Equations, 2023 (2023), no. 76, 1-20. [12] X. Chang, L. Jeanjean, N. Soave; Normalized solutions of L2-supercritical NLS equa- tions on compact metric graphs, Ann. Inst. H. Poincaré C Anal. Non Linéaire, DOI: 10.4171/AIHPC/8. [13] X. Chang, M. Liu, D. Yan; Normalized ground state solutions of nonlinear Schrödinger equa- tions involving exponential critical growth, J. Geom. Anal., 33 (2023), Paper No. 83, 20pp. [14] A. Fernández, L. Jeanjean, R. Mandel, M. Maris; Non-homogeneous Gagliardo-Nirenberg inequalities in RN and application to a biharmonic non-linear Schrödinger equation, J. Dif- ferential Equations, 330 (2022), 1-65. [15] G. Fibich, B. Ilan, G. Papanicolaou; Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. [16] B. A. Ivanov, A. M. Kosevich; Stable three-dimensional small-amplitude soliton in magnetic materials, So. J. Low Temp. Phys., 9 (1983), 439-442. [17] L. Jeanjean; Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. [18] L. Jeanjean, J. Jendrej, T. T. Le, N. Visciglia; Orbital stability of ground states for a Sobolev critical Schrödinger equation, J. Math. Pures Appl. (9), 164 (2022), 158-179. [19] L. Jeanjean, T. T. Le; Multiple normalized solutions for a Sobolev critical Schrödinger equa- tion, Math. Ann., 384 (2022), 101-134. [20] V. I. Karpman; Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. [21] V. I. Karpman, A. G. Shagalov; Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys D, 144 (2000), 194-210. [22] Y. Li, X. Chang, Z. Feng; Normalized solutions for Sobolev critical Schrödinger-Bopp- Podolsky systems, Electron. J. Differential Equations, 2023 (2023), no. 56, 1-19. [23] M. Liu, X. Chang; Normalized ground state solutions for nonlinear Schrödinger equations with general Sobolev critical nonlinearities, Discrete Contin. Dyn. Syst. Ser. S., DOI: 10.3934/dcdss.2024035. [24] T. Luo, S. Zheng, S. Zhu; The existence and stability of normalized solutions for a bi-harmonic nonlinear Schrödinger equation with mixed dispersion, Acta Math. Sci. Ser. B (Engl. Ed.), 43 (2023), 539-563. [25] X. Luo, T. Yang; Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient, Sci. China Math., 66 (2023), 1237-1262. [26] H. Lv, S. Zheng, Z. Feng, Existence results for nonlinear Schrödinger equations involving the fractional (p,q)-Laplacian and critical nonlinearities, Electron. J. Differential Equations, 2021 (2021), no. 100, 1-24. [27] Z. Ma, X. Chang; Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities, Appl. Math. Lett., 135 (2023), Paper 108388, 7pp. [28] Z. Ma, X. Chang, H. Hajaiej, L. Song; Existence and instability of standing waves for the biharmonic nonlinear Schrödinger equation with combined nonlinearities, arXiv.2305.00327. [29] C. Miao, G. Xu, L. Zhao; Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749. [30] F. Natali, A. Pastor; The fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347. [31] L. Nirenberg; On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13 (1959), 115-162. 20 Z. MA, X. CHANG, Z. FENG EJDE-2024/29 [32] B. Pausader, S. Xia; Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. [33] T. V. Phan; Blowup for biharmonic Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys., 69 (2018), Paper No. 31, 11pp. [34] N. Soave; Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), 6941-6987. [35] N. Soave; Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 108610, 43pp. [36] C. A. Swanson; The best Sobolev constant, Appl. Anal., 47 (1992), 227-239. [37] S. K. Turitsyn; Three-dimensional dispersion of nonlinearity and stability of multidimensional solitons, Teoret. Mat. Fiz., 64 (1985), 226-232. (in Russian) [38] J. Wei, Y. Wu; Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal., 283 (2022), Paper No. 109574, 46pp. [39] M. Willem; Minimax Theorems, Progr. Nonlinear Differential Equations Appl., 24. Birkhäuser Boston, Inc., Boston, MA, 1996. Zhouji Ma School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China Email address: mazj588@nenu.edu.cn Xiaojun Chang (corresponding author) School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, China Email address: changxj100@nenu.edu.cn Zhaosheng Feng School of Mathematical and Statistical Sciences, University of Texas Rio Grande Val- ley, Edinburg, TX 78539, USA Email address: zhaosheng.feng@utrgv.edu 1. Introduction and main results 2. Preliminary results 3. Case 2