1 Introduction

where H2= H2(Rd) := {v ∈L2| ?v ∈L2,?v ∈L2} is the standard Sobolev space which serves as the energy space.The BNLS type equation was introduced in [21, 22], where it took into account the role of a small fourth-order dispersion term in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity.The case μ = 0 was considered earlier, in[20, 34], in the context of the stability of solitons in magnetic materials when the effective quasi-particle mass becomes infinite.For a formal derivation of (1.1), the case μ > 0 arises in the approximation to the vectorial nonlinear Helmholtz equation as a nonparaxial correction to the second-order NLS; the case μ < 0 arises in the approximation to faster transmission propagation in optical fiber arrays (see [13] and the references therein).

Recently, the fourth-order nonlinear Schr¨odinger equations have received increasing attention.The local well-posedness for the Cauchy problem (1.1)–(1.2) in H2was obtained in[4, 23, 28].Fibich, Ilan and Papanicolaou[13] studied the global well-posedness for (1.1)–(1.2)in H2in the case where μ ≥0 and p ∈(1,1+8d].For if p ≥1+8d, Boulenger and Lenzmann[9]proved the existence of blowup solutions for(1.1)–(1.2);also see related numerical results in[2].This suggests that p = 1+8dis the critical exponent for the global existence and blowup for (1.1).The articles [26, 29, 31] studied the scattering for the fourth-order NLS.

In this paper we study the existence of standing wave solutions and their stability for(1.1).Let μ,ω ∈R and u=u(x) be a solution of the following elliptic equation:

In order to treat the case μ < 0 for the existence and stability of ground states for (1.3),we consider the minimization problem as

It is easy to see, for any u ∈Mμ, that there exists ω ∈R (namely, a Lagrange multiplier) such that (u,ω) solves eq.(1.3).Since u minimizes the energy Eμon B1, then u is a ground state solution (g.s.s.) of (1.3); see Theorem 6.1 in the Appendix.The definition of orbital stability for Mμis standard, and is given in Section 2.Note that the definition here refers to a weak type orbital stability, since it is about the stability of the set Mμrather than the stability of a single standing wave.

Then the set Mμ/=?and is orbitally stable.

We would like to mention that the results (1)–(3) of the theorem overlap somewhat with[5]and [12, 30].However,our approach is different.Applying mainly the profile decomposition method allows us to treat systematically all of the cases (1)–(4), which does not seem possible using the arguments in the above mentioned references.Moreover, we are able to extend our approach to prove the existence and stability, in Theorems 1.5 and 1.6, for negative μ in the mass-critical case p=1+.

Note that when μ=0, the problem (VP) reduces to the following:

In particular, u0∈B1is a minimizer of m0, where m0is given by (1.9).

The existence of Mμ,bis proven in Proposition 4.3, and consequently, the orbital stability follows in the same manner as in Theorem 1.1.Note that when μ <0 and b ≥b?, Lemma 4.2 shows that mμ,b= ?∞, meaning that (VP-b) is unsolvable.Also, Lemma 4.1 says that whenμ > 0, we have that Mμ,b= ?for all b > 0.The case μ = 0 admits ground state solutions if and only if b=b?exactly.

Theorem 1.6 can be proven nearly verbatim following the same proof as for Theorem 1.5,which supplements the results of Theorems 1.2 and 1.4 in [6] on the case μ > 0.Theorem 1.1 and Theorem 1.6 show that the sign of the second-order dispersion has a crucial effect on the construction of orbitally stable standing waves for the BNLS, especially when μ is negative.This is the case where ?2and ?μ?have played opposite roles for the dispersion of the energy that arises in physics [2, 9, 21, 22].Notably, in the mass critical case p = 1+8d, we find that when ?λ1<μ<0,the term ?μ?contributes to the existence of orbitally stable ground states for (1.1), while in the case μ > 0, there exist no ground states.Note that the result on (1.1)for μ=0 corresponds to the classical second order NLS,accounting for the L2-critical regimes,cf.[13, 35, 40].

We would like to mention that our proofs of the main theorems give a simple systematic method for showing the existence of g.s.s.that includes non-radial solutions for (1.1) based on the profile decomposition analysis.From Theorem 1.1 and Theorem 1.4, we see the upper bound μ0is sharp for μ>0 and p ∈[1+4d,1+8d).Moreover,the variational argument allows us to determine a lower bound of μ < 0 for p ∈(1,1+8d] regarding the existence of ground states for (VP) and (1.13).From the proofs, we conjecture that the lower bounds λ0and λ1in Theorem 1.1 and Theorem 1.6 are optimal, and are intrinsically dependent on the ground state of (1.3) with μ = 0.However, the uniqueness and symmetry problem seems to remain unsettled,other than knowing that ?θ,y ∈R,eiθQ(·?y)∈Mμfor all Q ∈Mμ.In this respect,the papers [5, 6] use the classical concentration-compactness method to study the existence of radially symmetric g.s.s.,however,the argument does not seem to directly apply to the focusing case μ<0.

For p > 1+4d, the orbital stability of the classical second-order NLS was considered in[36], and later, the result on NLS was significantly extended in [18] for general Hamiltonian systems that are invariant under a group of transformations.The analogous results for (1.1)in the L2-supercritical case were studied in [5, 6, 27] via a Lyapunov functional method for μ≥0 and p > 1+8d.The profile decomposition method has potential applications to the study of such a problem in the case where μ < 0 and p > 1+8d.The analysis in this paper can be further extended to address the orbital stability for higher-order Schr¨odinger type equations with potentials, based on the analogues for the NLS [15, 33, 37, 38] and related dynamical properties for general Hamiltonian partial differential equations near the standing waves [25,39, 42].

The remainder of this paper is organized as follows: in Section 2, we mainly state the local well-posedness of (1.1)–(1.2), the profile decomposition in H2, and a sharp Gagliardo-Nirenberg inequality for ?2.In Sections 3 and 4,we shall prove Theorem 1.1 and Theorem 1.5,respectively, with regard to the construction of ground states for (1.1).

2 Preliminaries

where on(1)→0 as n →+∞.

A primary advantage of the profile decomposition is the almost-orthogonality that can be used to defeat the lack of compactness of the given bounded sequence, as can be seen from the proof of Theorem 1.1 in Section 3.In solving the variational problem (VP), we also need the following sharp Gagliardo-Nirenberg type inequality obtained in, e.g., [13] and [40]:

It is worth mentioning that owing to Pohozaev identities and the scaling-invariance of the Jfunctional,any minimizer of the J-functional is a solution of (1.8)up to a scaling u(x)=βQ(αx)and vise versa,where Q is a solution of(1.8).Also,any J-minimizer is an E0-minimizer modular scaling and vise versa.

The Pohozaev type identity has a general version, as given in [17, Appendix 4.10], cf.[6, 9, 13] for some special cases.

To conclude this preliminary section, we recall the definition of the orbital stability for(1.1), which is stated in Theorems 1.1, 1.5 and 1.6.

Definition 2.6 The set Mμis said to be orbitally stable if, for any given ε > 0, there exists δ >0 such that, for any initial data ψ0satisfying

In other words, if the initial data ψ0is close to an orbit u ∈Mμ, then the corresponding solution ψ(t,x)of the system(1.1)–(1.2)remains close to the set of orbits Mμfor all time.The analogous definition applies to the orbital stability for Mμ,band Mcμin (1.11) and (1.13).

3 Construction of Standing Waves in the L2 Subcritical Case

by which we conclude that mμn→mμas μn→μ?.Similarly, we can show that mμn→mμas μn→μ+.This proves the continuity of mμfor all μ∈R.?

Lemma 3.2Let p>1.Then mμ≤0 for any μ∈R.

ProofLetting v0∈B1be fixed and defining the scaling for all ρ>0,

from which we conclude that mμ< 0 if μ > 0 is small enough.Thus, by the definition of μ0,we must have that μ0> 0.To prove μ0< ∞, it suffices to show that mμ= 0 for μ > 0 large enough.

For this purpose, we recall that in [5, (2.3)], the authors established the estimate

We are ready to prove the existence of a minimizer for(VP).We show that the the infimum of (VP) can be achieved by using the profile decomposition of bounded sequences in H2.

This implies, in view of the definition of mμ, the positivity of μ≥0, (3.26) and (2.5)

Thus, since {vn} is weakly convergent in H2, we must have limnvn(·+ xj0n) = Vj0in H2strongly.Therefore, we have proven Proposition 3.5.?

4 Construction of Standing Waves in the L2 Critical Case

Moreover, if μ > 0, then for all b ∈(0,b?], the functional Eμ,b(u) has no critical point on B1;that is,mμ,bcannot be achieved for any b>0.Ifμ=0,then a ground state solution of(VP-b)exists if and only if b=b?.

In the remainder of this section we mainly consider the case μ<0.

namely, Mμ,b/=?.

The proof of Proposition 4.3 is given by Lemmas 4.4 and 4.5, below.One main ingredient in the proof of the proposition is to establish the non-vanishing property for any minimizing sequence of (VP-b).

If p < 1+8d, μ < 0, one can prove (3.27) based on the fact that m0< 0; see (3.16) and(3.32).However, if p=1+8d, we know that m0,b=0 for all 0

Lemma 4.4Let μ<0 and 0

Remark 4.6In view of Lemma 4.2, the constant b?is a sharp upper bound in the sense that Mμ,b= ?for any b ≥b?.The analysis in the proof of Lemma 4.5 also seems to suggest that b?is a sharp lower bound, however,we are not able to verify this at present.

Proof of Proposition 4.3We note from the proof of Proposition 3.5, Case (ii), that to prove (4.6) and the fact that any minimizing sequence of mμ,bis pre-compact in H2by the profile decomposition method, we only need to show that any minimizing sequence of mμ,bis non-vanishing in the sense of (4.9).However, according to Lemmas 4.4 and 4.5, (4.9) holds under the assumptions on μ and b in this proposition.Therefore, we have proven Proposition 4.3.?

Proof of Theorem 1.5Let ψ(t)be the solution of the Cauchy problem(1.1)–(1.2)with initial datum ψ0∈H2.By (4.4) and the conservation laws of energy and mass in Proposition 2.1, we deduce that, for all t ∈I :=[0,T),

If 0 < b < b?, then (4.15) implies that {‖ψ(t)‖H2} is bounded for all t ∈I.Thus, from Proposition 2.1, we know that ψ(t) exists globally in time.In virtue of Proposition 4.3, it remains to show the stability of Mμ,bby a standard contradiction argument, as given in the proof of Theorem 1.1.Therefore, the proof is complete.?

Proof of Theorem 1.6As we remarked in the introduction, the proof of Theorem 1.6 is a straightforward technical translation taken verbatim from that of Theorem 1.5.?

5 Concluding Remarks

The study of stable ground states solutions has been a central problem for higher-order dispersive equations in the past few decades.Regarding the existence and stability theory for standing waves of fourth-order NLS (1.1) there has been much activity [2, 5, 6, 13, 27, 30, 32],especially, where the case μ ≥0 was mainly considered using different methods.Our primary contribution has been to treat the technically more challenging case μ < 0 by constructing an orbitally stable set of g.s.s.This has filled the gap in question.Moreover, at the critical exponent p = 1+8d, we have essentially shown in Theorem 1.6 and Lemma 4.2 that ‖Q?‖2is the threshold for the existence of the g.s.s.of (1.13), or equivalently (1.1), for suitable μ<0.

The existence of minimizers for certain negative μ was partially studied in [8] using a different method that was restricted to the submanifold {‖u‖p+1= 1} in H2, which is an equivalent of the Nehari manifold.However, the admissible values of the mass levels or the relation between the range limit of μ and the g.s.s.Qp=Q(p,d).Also, the stability issue was not available via the Nehari manifold method.The profile decomposition method we employed allowed us to address the existence problem for(1.1)and(1.3)for both signs ofμ;this has given a simpler approach than, e.g., [5], in the regime p ∈(1,1+8d].We believe that the analysis in this paper has provided certain optimal ranges for the parameter μ with both signs as shown in Theorems 1.1 and 1.6.The profile decomposition also allowed us to study the stability and instability problems on a deeper level in the regime p ≥1+8d.In this respect, one can find in[9] some closely related open questions for the case μ < 0, in particular at the threshold level Q?if p = 1+8d(compared with [36] for the corresponding paradigm for the classical NLS).We will continue to investigate this model in a sequel to this work.The variational analysis elaborated in this paper and [38, 41, 42] could potentially lead to sharper and more accurate descriptions of the asymptotic behaviors for the solitary waves by incorporating some of the spectral information for the associated linearized operators around the ground state down the path [1, 5, 6, 9, 27, 36].

6 Appendix

In this section, we give a detailed proof of the conclusion that a minimizer of mμis indeed a ground state solution of (1.3).Here, standardly, a ground state solution is a solution whose action functional Sω(u)has the least energy value among all non-trivial solutions of (1.3),where

Now we prove.

Theorem 6.1 Let u0∈B1is a minimizer of mμ.Then there exists ω0∈R such that u0is a ground state solution of (1.3) with ω =ω0.

Proof Since u0∈B1is a minimizer of mμ, then, by the standard Lagrange multiplier theory, there exists a Lagrange multiplier ω0∈R such that (u0,ω0) solves

By taking an infimum, this proves that d ≥Sω0(u0).Hence, it follows that Sω0(u0) = d, and the proof is completed.?

AcknowledgementsShijun Zheng would like to thank Atanas Stefanov and Kai Yang for helpful comments.