Hamiltonian elliptic systems: a guide to variational frameworks
Consider a Hamiltonian elliptic system of type
where is a power-type nonlinearity, for instance
having subcritical growth, and is a bounded domain of , . The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.
Key words and phrases:Hamiltonian elliptic systems, subcritical elliptic problems, qualitative properties, dual method, reduction by inversion, Lyapunov-Schmidt reduction, symmetry properties, multiplicity results, positive and sign-changing solutions, ground state solutions, strongly indefinite functionals.
- 1 Introduction
- 2 Direct approaches
- 3 The dual method
- 4 A reduction by inversion
- 5 A Lyapunov-Schmidt type reduction
- 6 More on the symmetry properties of solutions
- 7 Concentration phenomena
- 8 Multiplicity results in the spirit of the Symmetric Mountain Pass Lemma
- 9 Sign-changing solutions
Consider the problem
where the coupling between the two equations is made through a Hamiltonian of the form , and is a bounded domain, . In the literature these systems are usually referred to as elliptic systems of Hamiltonian type. It is also said that the equations are strongly coupled, in the sense that if and only if ; moreover, as we will see along this paper, many other properties are shared by the components of each solution pair.
The study of such system can be made through the use of variational methods. Unlike in the case of gradient systems where the choice of the energy functional associated to the problem is straightforward, in the case of Hamiltonian systems like (1.1) there are several variational approaches available, each one with its advantages and disadvantages. The aim of this paper is to give an overview of several of these variational frameworks emphasizing that even if almost all of them are suitable to obtain existence and multiplicity theorems, some of them are more adequate to derive certain qualitative properties of the solutions. We also review some of the recent literature, complementing and updating in this way the surveys [48, Section 3] and [98, Section 4] with only a few overlaps. For instance, one of our main interests consists in the variational characterization of ground state solutions and on Nehari type approaches. These topics are not covered in [48, 98]. We also emphasize that in comparison to [48, 98], we focus on the simplest case where is a sum of pure powers in order to grasp the main ideas and to avoid too much technicalities.
As we already stated, we will focus on the model case
so that the system becomes
The assumptions on the positive powers and will be discussed in a while. Formally, the equations in (1.1) are the Euler-Lagrange equations of the action functional
We will use the notation
to denote the quadratic part of the functional, while denotes the canonical inner product of . An important question is to decide in which space the functional should be defined. A first natural choice could be to work with . In order to define the functional in , we need to assume that
whereas the strict inequality, for , is required in order to get compactness properties. However, as was simultaneously observed in [50, 68], this is too restrictive; indeed, the correct notion of subcriticality associated to (1.1) is
while criticality corresponds to lying on the so called critical hyperbola:
Following the aforementioned papers, we can motivate this fact at least in two different ways. First, if , the system (1.3) reduces to the fourth order problem
whose critical exponent is given by , which is larger than . Observe that this is consistent with the choice of in (1.7). On the other hand, the critical hyperbola also arises in the generalized Pohoz̆aev identity due to Pucci and Serrin , Clément et al. , Mitidieri , van der Vorst , Peletier and van der Vorst ; in case of (1.3), this identity reads as
for every . By choosing first , one shows that the system (1.3) does not have positive solutions on star shaped domains if lies on or above the critical hyperbola, namely if
The previous arguments show that one should aim at working with satisfying (1.6). However, under such assumption, it may happen that for instance (for ) , and thus the action functional may not be well defined on . This fact is the first reason why, in the literature, several, though equivalent, variational approaches are considered.
One of the facts that will come out of our exposition is that the more general and meaningful notion of superlinearity is not , both nonlinearities superlinear, but rather
This allows for instance . Both the conditions and (1.6) are strongly related with the strong coupling in (1.3); the ideia is that one of the exponents can go “slightly” outside the interval , as long as the other one compensates it.
In the first sections of this paper, namely from Section 2 to Section 5, we overview several variational frameworks which have been used in the literature to deal with Hamiltonian systems. All these approaches can also be used under Neumann boundary conditions. Here, for simplicity, we have decided to deal only with homogeneous Dirichlet conditions like in (1.1). Let us describe the content of these sections. The bibliographic references to each method and result can be found in the corresponding section.
We start in Section 2 by reviewing two possible frameworks built directly in the functional (1.4). For this reason, we shall call them direct approaches. The first one consists in using the Sobolev spaces , for some suitable . For this is not a Hilbert space and hence this approach is rarely used to prove existence results. Nevertheless, it allows one to give a definition of ground state solution for every subcritical, and it is also useful when proving energy estimates. The second direct approach deals with fractional Sobolev spaces, with the definitive advantage of providing an Hilbertian framework.
It will become clear from the direct approaches that another difficulty when dealing with (1.4) is the fact that , its quadratic part, is strongly indefinite in the sense that it is positive and negative respectively in two infinite dimensional subspaces which split the function space in two. Moreover, (1.4) does not have a mountain pass geometry; in particular, the origin is not a local minimum. Instead, one has to rely in other linking theorems of more complicated nature. Another related issue is the fact that the usual Nehari manifold is not suitable to describe the ground state level.
One alternative to get rid of the indefinite character of (1.4) is to use the dual method, which we describe in Section 3. In an informal basis, the method consists in taking the inverse of the Laplace operator, rewriting the system as
and defining , , which leads to
The associated energy functional, defined in a suitable product of Lebesgue spaces, has a mountain pass geometry.
Finally, other possibility is to reduce the problem to a scalar one (cf. Section 4 and 5). In Section 4 we explain the reduction by inversion, which heuristically consists in taking and replacing it in the second equation of (1.3), leading to the single equation problem of higher order
This approach allows to deal with the sublinear case as well, and reduces the problem of ground state solutions to the easier study of finding solutions which achieve the best constant of a related Sobolev embedding.
In Section 5, for the case , we introduce a Nehari type manifold of infinite codimension in order to characterize with a minimization problem the ground state level. Moreover, by exploiting the properties of (1.4) on the pairs of type and , one can find, for each , a function so that the energy (1.4) calculated on once again displays a mountain pass geometry, and its critical points (in ) correspond to solutions of the original system. This approach can be thought as being a Lyapunov-Schmidt type reduction.
By using either the dual or the reduction by inversion method, one can prove in a relatively easy way positivity and symmetry properties for ground state solutions. However, it seems that this result does not follow easily with the other methods. Hence, one of the interesting things about all these approaches is that each method is more suitable to prove certain properties of the solutions. We illustrate this in a deeper way in the second part of the paper, from Section 6 on, where we survey some recent literature, highlighting for each stated result the most suitable framework. In Section 6 we combine the reduction by inversion approach with some arguments based on polarization of functions to prove symmetry properties of ground state solutions for two classes of systems. In particular, we solve an open problem, cf. [27, p. 451], about the radial symmetry of ground state solutions of a system posed on . For the so called Hénon-type system, we prove the foliated Schwarz symmetry of the ground state solution, as well as we present a result about symmetry breaking. Section 7 reviews the existing concentration results available for (1.1); there, the chosen method is the Lyapunov-Schmidt type reduction. After that, in Section 8 we show how to obtain infinitely many solutions (by three different methods: the two reductions and a Galerkin type method) in both the symmetric case (1.3) as well as in the perturbation from symmetry problem. Finally, the last section is about sign-changing solutions; Subsection 9.1 deals with a very recent result of existence and symmetry properties of least energy nodal solutions via dual method, while Subsection 9.2 is about the existence of infinitely many sign-changing solutions of (1.3) via the Lyapunov-Schmidt type reduction.
To sum up, one can say in conclusion that it seems that the direct approaches are harder to apply to (1.1) and have been less used in the past, mainly because it is hard to deal directly with the strongly indefinite functional (1.4). The dual method seems to be more adapted to prove sign and symmetry results; the reduction by inversion to prove sign, symmetry and multiplicity results, while the Lyapunov-Schmidt type reduction is useful in proving concentration and multiplicity results.
We finish this introduction by stressing that, although this paper is mainly a survey, it contains some original results, proofs, and computations that have not appeared elsewhere. As an example we refer to:
the proof that the standard Nehari manifold cannot be used to define the ground state level (Proposition 2.1);
comprehensive proofs of the several characterizations of least energy level in Section 5;
with respect to the existing bibliography, we prove the concentration results in Section 7 under more general assumptions on the nonlinearities.
Notations. We will denote the -norm by . We will always assume , except it is specifically mentioned, and define if ; otherwise.
2. Direct approaches
In this section we present two approaches built directly on the action functional (1.4). In Subsection 2.1 we use the spaces , while in Subsection 2.2 we deal with fractional Sobolev spaces. We also make some remarks concerning least energy solutions. In order to simplify the presentation, throughout this section we focus on the model case (1.2), so that the system in consideration is (1.3).
2.1. The framework
Having in mind the goal of finding a space in which (1.4) is well defined for lying bellow the critical hyperbola, following for instance [41, Section 1] (see also [28, Section 2]), we observe that for every ,
Therefore the quadratic part of (1.4) is well defined on the product , and if for some , one has the embeddings
the integral of the Hamiltonian in (1.4) is finite. Let us suppose without loss of generality that . One easily sees that the previous embeddings are continuous and compact whenever is such that
There are now two possibilities: either and we can take any suitable large , or and we can choose satisfying
under the mere assumption that
In this framework, a weak solution of (1.1) is a critical point of , i.e. a couple such that
for every . Observe that if we assume , then we can choose and is an agreeable framework.
It is clear that is not a local minimum of . Indeed, the quadratic part is indefinite since is positive definite whereas is negative definite for . This implies that the functional does not display a mountain pass geometry. Moreover, is not a Hilbert space if , which makes linking theorems as the one by Benci and Rabinowitz  not applicable (we refer to the next subsection for a different framework which allows the use of linking theorems). Due to this fact, it is quite involved to show existence results using directly the functional . We refer to  or [98, Section 5] for an approach in that direction. On the other hand, once one knows that a solution actually exists (for example through other approaches), one can then use to obtain energy estimates. In [28, Section 2], for instance, this framework has proved itself to be useful in estimating the level of least energy solutions, also called ground state solutions. These can be defined as pairs achieving
A priori this level could depend on , but this turns out not to be the case. Indeed, arguing as in [27, Proposition 2.1], see also [101, Theorem 1] or Subsection 5.3 ahead, one shows with a standard bootstrap that the weak solutions of (1.1) are classical solutions, so that the numbers are independent of the particular choice of . Throughout this paper we will denote the ground state level simply by .
In the case of a single equation or when dealing with gradient systems, one possible characterization of the ground state level is through the minimization of the energy functional on the so called Nehari manifold. For Hamiltonian systems, this turns out to be unsuccessful, and we illustrate this fact in the superlinear case .
If is a weak solution of (1.3), we have . Moreover, using the fact that , we infer that
Assuming , we deduce that for a nontrivial weak solution of (1.3), we have
If (H1) holds, we have all the required compactness to prove that is achieved as soon as one can prove that has at least one critical point and we therefore deduce that .
Next, we define the Nehari manifold as usual by
In contrast with the case of a single equation or gradient systems, the origin turns out to be adherent to . This means , and therefore cannot be a critical level associated to a nontrivial critical point! Since this fact seems not so well known by the community and has been misused, we state it as a proposition for completeness.
Assume (H1) and hold. Then is an adherent point of and .
Suppose for instance that . Let be a positive function, and . Then means
Observe that as , whatever and are fixed.
Claim : for each large enough, there exists a unique such that
If , then for each we have . Thus the claim follows easily from the continuity of . For one can argue in an analogous way for each satisfying , since in such case . Finally, when , we can take such that
which leads to , and we conclude as before.
Conclusion : the identity (2.3) implies in particular that
Hence , and as , so that the proof is complete.
In Section 5, we will define a suitable Nehari type set (of infinite codimension). Namely, by imposing the relations
for every direction , we will recover that the minimum on such a set corresponds to the ground energy level. A different route will also be considered in Sections 3 and 4 where we provide two other ways to recover a characterization of the ground state level as the minimum on a standard Nehari manifold.
2.2. Using fractional Sobolev spaces
In this section we describe the variational approach based on the use of fractional Sobolev spaces, following [50, 68]. We recall that, in order to simplify the computations, we still assume , and refer to the above mentioned papers for more general statements. The following approach will yield an existence result for such that
Recall that is equivalent to , which corresponds to the notion of superlinearity in the context of elliptic Hamiltonian systems.
(both nonlinearities need to be superlinear), while the observation that one can actually treat the more general case (H2) is done in . For more precise details check the proof of Theorem 2.4 ahead.
For , the fact that lies below the critical hyperbola may yield that (for instance) . In such a case, we cannot define the action functional (1.4) in , and the idea is to impose a priori more regularity on and less on , keeping at the same time an Hilbertian framework. Having this in mind, let us introduce the fractional Sobolev spaces .
Let be the sequence of –normalized eigenfunctions of , with corresponding eigenvalues . It is well know that each coincides with its Fourier series
with . For , we can therefore define the operator , where
We endow with the inner product
so that is an Hilbert space with the Hilbertian norm . We denote by the inverse of the operator . Observe that and , while .
Suppose for the moment that
whence we can take such that
This last statement is equivalent to
Thus, under this choice, we have the compact embeddings (see [50, Theorem 1.1]):
To simplify the notation, we set and . The previous embeddings imply that the energy functional
is a well defined –functional for as in (2.5). Then is a critical point of if and only if
for every that is, is a solution of
This is the notion of weak solution in this context. It is proved in [50, Theorem 1.2] that weak solutions are strong solutions, in the sense that
and they satisfy the system (1.1) pointwise for a.e. . By using a bootstrap argument and elliptic regularity theory , see also Subsection 5.3 ahead, one proves in a standard way that weak solutions are in fact classical solutions.
In order to obtain the existence of nontrivial solutions under (H2) we need some preliminaries. First observe that the functional may be written in the form
where is the self-adjoint bounded linear operator defined by the condition
having the explicit formula
The space decomposes in , with
Observe that both and are infinite dimensional, and the quadratic part is positive on , negative on . In the literature, this type of geometry is referred to as strongly indefinite.
Under (H2), the functional satisfies the Palais-Smale condition.
Let be so that is bounded and . Then we have
with as . Thus
Since is bounded in , up to a subsequence, weakly converges to some . The convergence is actually strong and this can be deduced from a careful analysis of the convergence
Sketch of the proof.
We follow , where more general nonlinearities are considered.
Step 1. Definition of the set . Given , by using the embeddings (2.1), we have that, for ,
for some independent of . Define the set
Then there exists a constant such that whenever is taken sufficiently small.
Step 2. Definition of the set . Let be any eigenfunction of . Given constants , define the set
and , it can be proved that for sufficiently large (cf. [50, Section 3]).
Step 3. Conclusion. It can be proved that and link111In the sense of equation (3.7) in ., and one can apply the linking theorem of Benci and Rabinowitz [23, Theorem 0.1] in a version due to Felmer [61, Theorem 3.1] (see also  where, under the additional assumption that , [23, Theorem 0.1] is applied directly with , ). ∎
Once we have the existence of at least one solution, the existence of a solution with least energy follows from a compactness argument.
Take satisfying (H2). Then the ground state level
is achieved and positive.
First, observe that arguing as in the previous subsection, if and , we obtain
Hence we infer that . Suppose without loss of generality that , so that does hold. From the identity , we infer that
for some . In particular, there exists such that for every nontrivial solution ,
Now take a minimizing sequence at the level , that is