- Open Access
Gradient systems with sublinear term near the origin and asymptotically linear term near infinity
© Cai and Su; licensee Springer. 2013
Received: 24 September 2013
Accepted: 3 December 2013
Published: 30 December 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.
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
where if and if .
for . The weak solutions to (GS) in E are exactly critical points of Φ in E.
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.
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
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.
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.
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.
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 .
is called the q th critical group of Φ at , where denotes a singular relative homology group of the pair with integer coefficients.
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 .
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.
- (1)If , then
- (2)If , then
provided Φ satisfies the angle conditions with respect to :
Next we recall an abstract critical point theorem built by Wang in .
Proposition 2.3 ()
where , then Φ has a sequence of critical values satisfying as .
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.
Proof Denote . By definition of critical groups, we can write . We will construct a deformation mapping from to for small.
This contradicts (3.9). Thus and (3.8) holds.
The proof is complete. □
We note here that () is not comparable with () since () is a local condition and although () implies ()(i) but ()(ii) is a global condition.
The functional Φ is coercive on E and satisfies the Palais-Smale condition.
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.
The proof is complete. □
Φ satisfies the Cerami condition.
if () holds.
if () holds.
This is a contradiction with (3.25).
(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.
By (4.2) and (4.3), we see that and then is a nontrivial weak solution of (GS). □
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.
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.
() is weaker than ().
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 .
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.
Supported by NSFC11271264, NSFC11171204 and PHR201106118.
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