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Gradient systems with sublinear term near the origin and asymptotically linear term near infinity
Boundary Value Problemsvolume 2013, Article number: 280 (2013)
In this paper we study the existence of nontrivial solutions for a sublinear gradient system with a nontrivial critical group at infinity.
MSC:35J10, 35J65, 58E05.
In this paper, we are concerned with the following gradient system
where is a bounded open domain with a smooth boundary ∂ Ω and designates the partial derivative with respect to u of the nonlinearity . The solutions of such systems are steady-states of reaction-diffusion systems arising in many applied sciences such as biology, chemistry, ecology or physics. It is well known that (GS) has variational structure when the nonlinearity F satisfies the subcritical growth condition
(F) and there are and such that
where if and if .
That is, the solutions of (GS) can be found as critical points of the following functional
defined on which is a Hilbert space endowed with the inner product
and the associated norm
By the compact Sobolev embedding with , under the global assumption (F), the functional Φ is well defined and is of class (see ) with its Fréchet derivative
for . The weak solutions to (GS) in E are exactly critical points of Φ in E.
We make some conventions. We use and to denote the norm and the inner product in and use to denote an element in and E. Bz denotes the matrix product in for a matrix B and . We use 0 to denote the origin in various spaces. Let be the set of all continuous, cooperative and symmetric matrix functions on . A matrix function takes the form
with the functions satisfying the conditions that for all , which means A is cooperative, and that .
When F satisfies , for , the system (GS) admits a trivial solution . We are interested in the nontrivial solutions for (GS). In the current paper we apply the Morse theory to study the existence of nontrivial solutions of (GS) when the problem is sublinear near the origin and is asymptotically linear near infinity.
We make the following assumption near the origin.
() , and there are and such that
In order to state the assumptions on the nonlinearity at infinity, we need some basic facts about the eigenvalue problem of linear gradient system. For a given matrix , it is known (see [2, 3]) that the corresponding linear system
admits a sequence of distinct eigenvalues of finite multiplicity
such that as . According to A, the space E can be split as
The numbers , are well determined and finite.
We assume that the nonlinear system (GS) is asymptotically linear at infinity in the sense that the function F satisfies
() there is a matrix such that
Associated to , we set , , . Denote , . We say that the system (GS) is nonresonant at infinity if , while it is resonant at infinity if .
We first consider the nonresonance case. We have the following.
Theorem 1.1 Assume that F satisfies (), () and . Then (GS) has at least one nontrivial weak solution in E.
Next we consider the resonance case. We need additional assumptions on F near infinity.
() for some . For , where , , and imply that there exist and such that
Theorem 1.2 Let F satisfy (), () and (). Then (GS) has at least one nontrivial weak solution in E. Moreover, if F is even in z, then (GS) has infinitely many nontrivial weak solutions in E.
Theorem 1.3 Let F satisfy (), () and (). Then (GS) has at least one nontrivial weak solution in E.
Now we give some remarks and comments. The gradient system represents the steady-state case of reaction-diffusion system which is a model for problems arising from biology, chemistry, physics and ecology, etc. In this paper we look for nontrivial solutions for the system (GS) via Morse theory. When the problem is resonant at infinity, we impose on the nonlinearity F the global assumption () to ensure the compactness and clear description of critical groups for Φ at infinity. () can be regarded as a variant of the famous Landesman-Lazer type resonance condition  which can be formulated as
See  for details. Near the origin we impose (), which means that ∇F is sublinear or F is sub-quadratic near zero. This kind of condition caught our attention first in a preprint by Liu and Wu  where a single elliptic equation was considered. This is the first use for gradient system in the current paper.
The asymptotically linear gradient systems (GS) have received some attention for years. We mention some recent related works [7–12] and the references therein. In these works, existence and multiplicity of nontrivial solutions for (GS) were obtained by combining various arguments involving Morse theory, saddle point reduction method (see [9–11]) and three critical point theorem (see ), etc. All above mentioned works dealt with the case that at least one of the critical groups of Φ at 0 is nontrivial somewhere. In the present paper, we study via Morse theory the case that all critical groups of Φ at 0 are trivial under the condition (). Due to (), the saddle point reduction methods [9–11] cannot be applied and there is no linking at 0. Comparing with known ones, the existence and multiplicity results for (GS) are all new. See more remarks in the last section of the paper.
The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3 we compute the critical groups at zero and infinity. The proofs of Theorems 1.1-1.3 and comments are given in Section 4.
In this section we cite some preliminaries that will be used to prove the main results of the paper. We first collect some results on Morse theory (see [14, 15]) for a functional Φ defined on a Hilbert space E.
Let . Denote for
We say that Φ possesses the deformation property at the level if for any and any neighborhood of , there are and a continuous deformation such that
for either or ;
is non-increasing in t for any ;
We say that Φ possesses the deformation property if Φ possesses the deformation property at each level .
In applications the deformation property is ensured by the Palais-Smale condition or the Cerami condition.
We say that Φ satisfies the Palais-Smale condition at the level if any sequence satisfying and as has a convergent subsequence. Φ satisfies the Palais-Smale condition if Φ satisfies the Palais-Smale condition at each . We say that Φ satisfies the Cerami condition [16, 17] at the level if any sequence satisfying that , as has a convergent subsequence. Φ satisfies the Cerami condition if Φ satisfies the Cerami condition at each .
Let be an isolated critical point of Φ with , and U be a neighborhood of . The group
is called the q th critical group of Φ at , where denotes a singular relative homology group of the pair with integer coefficients.
Let . Assume that is bounded from below by and Φ possesses the deformation property at all . Then the group
is called the q th critical group of Φ at infinity .
Assume that Φ satisfies the deformation property and is a finite set. The Morse type numbers of the pair are defined by , and the Betti numbers of the pair are defined by .
Proposition 2.1 Assume that possesses the deformation property, , and all , are finite and only finitely many of them are nonzero. Then it holds
If , then for all . From (2.1) one can deduce that for all . Thus if for some , then Φ must have a critical point with . If , then for all . Thus if for some , then Φ must have a new critical point. Therefore the basic idea in applying Morse theory to find critical points of Φ is to compute critical groups both at infinity and at known critical points clearly and then to find unknown critical points by applying formulas (2.1) and (2.2).
Now we state an abstract result for the critical groups at infinity.
Proposition 2.2 Let the functional take the form
where is a self-adjoint linear operator such that 0 is isolated in , the spectrum of ℒ. Assume that satisfies
Denote , , where are subspaces on which ℒ is positive (negative) definite. Assume that and are finite, and Φ possesses the deformation property.
If , then
If , then
provided Φ satisfies the angle conditions with respect to :
() there exist and such that
Next we recall an abstract critical point theorem built by Wang in .
Proposition 2.3 ()
Let , where X is a Banach space. Assume that Φ possesses the deformation property, is even and bounded from below, and . If for any , there exist k-dimensional subspaces and such that
where , then Φ has a sequence of critical values satisfying as .
Finally, we mention the eigenvalues of the linear gradient system (LA). By the compact embedding , for a given , there is a compact self-adjoint operator associated with A such that
The operator possesses the property that is an eigenvalue of (LA) if and only if there is nonzero such that
(LA) has a sequence of distinct eigenvalues
and each eigenvalue of (LA) has a finite multiplicity. All eigenvectors of (LA) form a Hilbertian basis of E and that E can be split as , where , , are the negative, positive definite invariant subspaces and the kernel of , respectively. We refer to [2, 3] for more properties related to the eigenvalue problem (LA) and the operator .
3 Critical groups and compactness
In this section we verify the compactness of the functional Φ and compute the critical groups of Φ at both zero and infinity. Without loss of generality, we assume that (GS) has finitely many weak solutions so that the trivial solution is an isolated critical point of Φ. We first compute the critical groups . The idea was from an unpublished preprint by Liu and Wu  where a single elliptic equation was studied.
We work with the functional
Lemma 3.1 Assume that F satisfies (F) and (), then
Proof Denote . By definition of critical groups, we can write . We will construct a deformation mapping from to for small.
In the following we use to denote positive constants. By (), one deduces that
It follows from (F) and (2.2) that for some ,
For and , we have
Since , for each given , there exists such that
Let be such that and . Then
From (3.6), one concludes that there exists such that
From now on we fix . We claim that
Let and . By the continuity of Φ, there exists such that for all . We will get (3.8) by proving . Suppose that . Then there is some such that
As , it follows from (3.7) that
But implies that
This contradicts (3.9). Thus and (3.8) holds.
Now define a mapping as
By (3.5), (3.7) and (3.8), for with , there exists a unique such that
Thus the mapping π is well defined. Moreover, it follows from (3.7), (3.12) and the implicit function theorem that the mapping π is continuous in z. Define a mapping by
Then η is a continuous deformation from to . By homotopy invariance of a homology group and the contractibility of , we have
The proof is complete. □
() there are and such that
We note here that () is not comparable with () since () is a local condition and although () implies ()(i) but ()(ii) is a global condition.
Now we verify the compactness for the functional Φ and compute the critical groups of Φ at infinity. To do this, we rewrite the functional Φ as
Lemma 3.2 Let F satisfy () and ().
The functional Φ is coercive on E and satisfies the Palais-Smale condition.
Proof (i) First, () implies (F) while () implies
We will prove that
Assume that there is a sequence such that for some , it holds
Set . Then for all . Up to a subsequence, we may assume that there is such that
By (3.13) one has that for some constant ,
Therefore by (3.14) we deduce that
Taking in (3.18), it follows from (3.15) and (3.16) that
On the other hand, we have by the lower semi-continuity of the norm that
By (3.16), (3.21) we get
Hence and is an eigenvector corresponding to the first eigenvalue . It follows that for almost every and then
Now it follows from (3.13), (3.23) and the Fatou lemma that
This is a contradiction. Thus Φ is coercive on E.
By the coercivity of Φ, a Palais-Smale sequence of Φ must be bounded. Since F has a subcritical growth, a standard argument shows that has a convergent subsequence.
(ii) Since Φ is coercive and weakly lower semi-continuous, Φ is bounded from below. Take . Then
The proof is complete. □
Lemma 3.3 Let F satisfy () and ().
Φ satisfies the Cerami condition.
if () holds.
if () holds.
Proof (i) Let be such that
We only need to show that is bounded in E. Suppose, by the way of contradiction, that
Denote , then . Passing to a subsequence if necessary, we may assume that there is such that as ,
By () and (3.26) we deduce that is bounded in and
For , we have
Letting , it follows that
Taking in (3.29) and using (3.25), (3.28), we obtain that
Taking in (3.30), we obtain that
Therefore and thus
and is an eigenvector of (L) associated with eigenvalue 1. It follows that . Write , where , , then
By () there exist and such that
This implies that
This is a contradiction with (3.25).
(ii) We apply Proposition 2.2. Set and
Then Φ can be rewritten as
By Lemma 3.1, Φ satisfies the Cerami condition and hence possesses the deformation property. From () one sees that satisfies
Now we show that Φ satisfies the angle condition () at infinity with respect to the orthogonal decomposition when () holds. Suppose it is not true, then for each , there are , , , such that
It follows from (3.35) that
By () we have that for ,
this contradicts (3.36). Therefore () holds, and by Proposition 2.2 we have
(iii) This case is proved in a similar way.
The proof is finished. □
4 Proofs of main theorems
In this section we give the proofs of main theorems in this paper.
Proof of Theorem 1.1 By (), the functional Φ takes the form
where is a bounded self-adjoint linear operator, with a compact gradient satisfying
Since , the problem is no resonance at infinity and thus Ψ satisfies the Palais-Smale condition. By Proposition 2.2(1) we have
Therefore Φ has a critical point satisfying
By Lemma 3.1 we have
By (4.2) and (4.3), we see that and then is a nontrivial weak solution of (GS). □
Proof of Theorem 1.2 By (), () and Lemma 3.2, we have
Therefore Φ has a critical point satisfying
We still have (4.3). Thus is a nontrivial weak solution of (GS). In fact, is a global minimizer of Φ.
Assume that is even in z. We will employ Proposition 2.3 to prove the multiplicity in Theorem 1.2. Now Φ is even, . By Lemma 3.2, Φ satisfies the Palais-Smale condition and is bounded from below following from the coercivity.
We verify (2.5). Let be a k-dimensional subspace of E. For , as arguments in the proof of Lemma 3.1, we have
Since and all norms on are equivalent, we get that for small enough,
With all the conditions of Proposition 2.3 being verified, we get the conclusion that Φ has a sequence of critical values satisfying as . Thus (GS) has infinitely many nontrivial weak solutions in E. The proof is finished. □
Proof of Theorem 1.3 By a similar argument, it follows from Lemma 3.1 and Lemma 3.3. □
We conclude the paper with further comments and remarks.
Remark 4.1 (i) In Theorem 1.1, when which implies and F is even in z, by the same arguments as the last part of the proof of Theorem 1.2, one can show that (GS) has infinitely many nontrivial weak solutions in E with negative energies which converge to zero.
(ii) In Theorem 1.2, one nontrivial solution could be obtained if () is replaced by the nonquadraticity condition 
Indeed, () is equivalent to
which implies (), i.e.,
() is weaker than ().
(iii) The result for one nontrivial solution in Theorem 1.2 is valid when F satisfies (), () and the nonquadraticity condition 
Indeed, in this case, Φ satisfies the Cerami condition and Φ has a saddle point structure at infinity with respect to in the sense that Φ is bounded from below on and is anti-coercive on . Then Proposition 3.8 in  is applied to get .
Remark 4.2 In Theorem 1.2, we proved the multiplicity result by a critical point theorem in  when Φ is even. This result is completely new for gradient systems. Since the critical groups of Φ at both zero and infinity are clearly computed, when Φ is even, the Morse equality may provide us an idea to give a different proof provided we have in hand the following basic conclusion.
(⋆) If is a solution of (HS), then for finitely many .
Let (⋆) hold and let F be even. We prove the multiplicity for (GS) in Theorem 1.2 via Morse theory. Assume that (GS) has only finitely many pairs of nontrivial solutions. Denote . Then by the Morse equality, one has that
By (⋆), (4.3) and (4.4), it follows that
a contradiction. Similarly, if (⋆) is valid and F is even, then we have the same multiplicity result in Theorems 1.1 and 1.3.
We note here that the conclusion (⋆) is valid for Φ is of . A natural problem arises here whether or not that (⋆) is valid for a functional. It is still open to the best of our knowledge. We will focus on this problem in near future.
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Supported by NSFC11271264, NSFC11171204 and PHR201106118.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.