Abstract
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.
Keywords:
gradient system; sublinear; critical group; Morse theory1 Introduction
In this paper, we are concerned with the following gradient system
where
(F)
where
That is, the solutions of (GS) can be found as critical points of the following functional
defined on
and the associated norm
By the compact Sobolev embedding
for
We make some conventions. We use
with the functions
When F satisfies
We make the following assumption near the origin.
(
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
admits a sequence of distinct eigenvalues of finite multiplicity
such that
where
The numbers
We assume that the nonlinear system (GS) is asymptotically linear at infinity in the sense that the function F satisfies
(
Associated to
We first consider the nonresonance case. We have the following.
Theorem 1.1Assume thatFsatisfies (
Next we consider the resonance case. We need additional assumptions on F near infinity.
(
(
Theorem 1.2LetFsatisfy (
Theorem 1.3LetFsatisfy (
Now we give some remarks and comments. The gradient system represents the steadystate
case of reactiondiffusion 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 (
See [5] for details. Near the origin we impose (
The asymptotically linear gradient systems (GS) have received some attention for years.
We mention some recent related works [712] 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 [911]) and three critical point theorem (see [13]), 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 (
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 1.11.3 and comments are given in Section 4.
2 Preliminary
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
Let
We say that Φ possesses the deformation property at the level
(1)
(2)
(3)
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 PalaisSmale condition or the Cerami condition.
We say that Φ satisfies the PalaisSmale condition at the level
If Φ satisfies the PalaisSmale condition or the Cerami condition, then Φ possesses the deformation property [14,16].
Let
is called the qth critical group of Φ at
Let
is called the qth critical group of Φ at infinity [18].
Assume that Φ satisfies the deformation property and is a finite set. The Morse type numbers of the pair
Proposition 2.1Assume that
If
Now we state an abstract result for the critical groups at infinity.
Proposition 2.2Let the functional
where
Denote
(1) If
(2) If
provided Φ satisfies the angle conditions with respect to
(
Proposition 2.1(1) was obtained in [19] (see Remark 5.2 in [14]). Proposition 2.1(2) is a revision of Proposition 3.10 in [18] which was made first in [20] and was remade in [21].
Next we recall an abstract critical point theorem built by Wang in [22].
Proposition 2.3 ([22])
Let
where
Finally, we mention the eigenvalues of the linear gradient system (L_{A}). By the compact embedding
The operator
(L_{A}) has a sequence of distinct eigenvalues
and each eigenvalue
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
We work with the functional
Lemma 3.1Assume thatFsatisfies (F) and (
Proof Denote
In the following we use
It follows from (F) and (2.2) that for some
For
Since
Let
From (3.6), one concludes that there exists
From now on we fix
Let
As
But
This contradicts (3.9). Thus
Now define a mapping
By (3.5), (3.7) and (3.8), for
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
Then η is a continuous deformation from
The proof is complete. □
We remark here that in [23] the similar idea for computing the critical groups at 0 was presented for a single elliptic equation. For (GS), the conditions used in [23] can be formulated as
(
We note here that (
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
where
Lemma 3.2LetFsatisfy (
(i) The functional Φ is coercive onEand satisfies the PalaisSmale condition.
(ii)
Proof (i) First, (
We will prove that
Assume that there is a sequence
Set
By (3.13) one has that for some constant
Therefore by (3.14) we deduce that
Taking
On the other hand, we have by the lower semicontinuity of the norm that
Thus
By (3.16), (3.21) we get
Hence
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 PalaisSmale sequence
(ii) Since Φ is coercive and weakly lower semicontinuous, Φ is bounded from below.
Take
The proof is complete. □
Lemma 3.3LetFsatisfy (
(i) Φ satisfies the Cerami condition.
(ii)
(iii)
Proof (i) Let
We only need to show that
Denote
By (
Therefore
For
Letting
Taking
Taking
Therefore
and
By (
This implies that
This is a contradiction with (3.25).
(ii) We apply Proposition 2.2. Set
Then Φ can be rewritten as
By Lemma 3.1, Φ satisfies the Cerami condition and hence possesses the deformation
property. From (
Now we show that Φ satisfies the angle condition (
and
It follows from (3.35) that
By (
this contradicts (3.36). Therefore (
(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 (
where
Since
Therefore Φ has a critical point
By Lemma 3.1 we have
By (4.2) and (4.3), we see that
Proof of Theorem 1.2 By (
Therefore Φ has a critical point
We still have (4.3). Thus
Assume that
We verify (2.5). Let
Since
With all the conditions of Proposition 2.3 being verified, we get the conclusion
that Φ has a sequence of critical values
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
(ii) In Theorem 1.2, one nontrivial solution could be obtained if (
Indeed, (
which implies (
(
(iii) The result for one nontrivial solution in Theorem 1.2 is valid when F satisfies (
Indeed, in this case, Φ satisfies the Cerami condition and Φ has a saddle point structure
at infinity with respect to
(iv) In Theorem 1.3, the global condition (
Remark 4.2 In Theorem 1.2, we proved the multiplicity result by a critical point theorem in [22] 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
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
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
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
Supported by NSFC11271264, NSFC11171204 and PHR201106118.
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