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 theory
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
That is, the solutions of (GS) can be found as critical points of the following functional
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
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
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.
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
We assume that the nonlinear system (GS) is asymptotically linear at infinity in the sense that the function F satisfies
We first consider the nonresonance case. We have the following.
Next we consider the resonance case. We need additional assumptions on F near infinity.
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.
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 qth critical group of Φ at infinity .
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.
Next we recall an abstract critical point theorem built by Wang in .
Proposition 2.3 ()
(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
The proof is complete. □
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
(i) The functional Φ is coercive onEand satisfies the Palais-Smale condition.
We will prove that
Therefore by (3.14) we deduce that
On the other hand, we have by the lower semi-continuity of the norm that
By (3.16), (3.21) we get
Now it follows from (3.13), (3.23) and the Fatou lemma that
This is a contradiction. Thus Φ is coercive on E.
The proof is complete. □
(i) Φ satisfies the Cerami condition.
This implies that
This is a contradiction with (3.25).
Then Φ can be rewritten as
It follows from (3.35) that
(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 Lemma 3.1 we have
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.
(ii) In Theorem 1.2, one nontrivial solution could be obtained if () is replaced by the nonquadraticity condition 
(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.
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.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and read and approved the final manuscript.
Supported by NSFC11271264, NSFC11171204 and PHR201106118.
Chang, KC: An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solutions problems. Acta Math. Sin.. 15, 439–454 (1999). Publisher Full Text
Chang, KC: Principal eigenvalue for weight in elliptic systems. Nonlinear Anal.. 46, 419–433 (2001). Publisher Full Text
Bartsch, T, Chang, KC, Wang, Z-Q: On the Morse indices of sign changing solution of nonlinear elliptic problem. Math. Z.. 233, 655–677 (2000). Publisher Full Text
Furtado, MF, de Paiva, FOV: Multiplicity of solutions for resonant elliptic systems. J. Math. Anal. Appl.. 319, 435–449 (2006). Publisher Full Text
Furtado, MF, de Paiva, FOV: Multiple solutions for resonant elliptic systems via reduction method. Bull. Aust. Math. Soc.. 82, 211–220 (2010). Publisher Full Text
Lü, L, Su, J: Solutions to a gradient system with resonance at both zero and infinity. Nonlinear Anal.. 74, 5340–5351 (2011). Publisher Full Text
Liu, JQ, Su, J: Remarks on multiple nontrivial solutions for quasi-linear resonant problems. J. Math. Anal. Appl.. 258, 209–222 (2001). Publisher Full Text
Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to nonlinear problems with ‘strong’ resonance at infinity. Nonlinear Anal.. 7, 981–1012 (1983). Publisher Full Text
Bartsch, T, Li, S: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal.. 28, 419–441 (1997). Publisher Full Text
Wang, Z-Q: Multiple solutions for indefinite functionals and applications to asymptotically linear problems. Acta Math. Sin. New Ser.. 5(2), 101–113 (1989). Publisher Full Text
Su, J: Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity. J. Differ. Equ.. 145(2), 252–273 (1998). Publisher Full Text
Su, J, Zhao, L: An elliptic resonance problem with multiple solutions. J. Math. Anal. Appl.. 319, 604–616 (2006). Publisher Full Text
Wang, Z-Q: Nonlinear boundary value problems with concave nonlinearities near the origin. Nonlinear Differ. Equ. Appl.. 8, 15–33 (2001). Publisher Full Text
Costa, DG, Magalhães, CA: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal.. 23, 1401–1412 (1994). Publisher Full Text
Su, J, Tang, C: Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues. Nonlinear Anal.. 44, 311–321 (2001). Publisher Full Text