- Open Access
Asymptotically linear systems near and at resonance
© Chhetri and Girg; licensee Springer 2014
- Received: 20 June 2014
- Accepted: 4 November 2014
- Published: 20 November 2014
This paper deals with an elliptic system of the form in Ω, in Ω, on ∂ Ω, where is a parameter and () is a bounded domain with -boundary ∂ Ω, (a bounded open interval if ). Here with a.e. in Ω and are constants. The nonlinear perturbations are Carathéodory functions that are sublinear at infinity. We provide sufficient conditions for determining the λ-direction to which a continuum of positive and negative solutions emanates from infinity at the first eigenvalue of the associated linear problem. Furthermore, as a consequence of main results, we also provide sufficient condition for the solvability of a class of asymptotically linear system near and at resonance satisfying Landesman-Lazer type conditions.
- elliptic systems
- asymptotically linear
- bifurcation from infinity
- Landesman-Lazer condition
where () is a bounded domain with -boundary ∂ Ω, (a bounded open interval if ). We will assume that is a parameter, are constants, a.e. in Ω is an function. The nonlinear perturbations satisfy the following assumptions uniformly on compact intervals of λ:
(H1) f and g are Carathéodory functions;
Therefore, the system (1.1) is asymptotically linear at infinity.
for , endowed with the norm . We say that is a solution of (1.1) if solves (1.1) in the strong sense, that is, and satisfies (1.1) almost everywhere in Ω. Further, if (<0) and (<0) almost everywhere in Ω then we say that is a positive (negative) solution of (1.1).
By a continuum of solutions of (1.1) we mean which is closed and connected.
A continuum ℭ bifurcates from infinity at if there exists a sequence of solutions such that and as .
It is the only eigenvalue such that both components of its eigenfunction, , are positive (or negative) in Ω. Further, for each , denotes the eigenvalue of (1.4). For a detailed discussion of the eigenvalue problem (1.4), see Appendix 2.
Now we state our main results.
Let (H1)-(H3) hold. Then there existsuch that any solutionwithandsatisfyandorand, andfor anywith. Moreover, there are continuabifurcating from infinity atcontaining positive and negative solutions, respectively.
Theorem 1.1 establishes the existence of positive and negative solutions of (1.1) near . In the next two theorems we determine the λ-direction to which the continua from Theorem 1.1 bifurcate. For this we impose additional conditions on the perturbations f and g below.
Now let ε, K, , be as in Theorem 1.1. Then we prove the following results.
there is no positive (negative) solution satisfying and , and
the continuum , consisting of positive (negative) solutions, bifurcates from infinity at to the left.
there is no positive (negative) solution satisfying and , and
the continuum , consisting of positive (negative) solutions, bifurcates from infinity at to the right.
This paper is motivated by the results obtained for the scalar case in , Theorem 3 and Theorem 4] and in , Theorem 1]. The goal of the present paper is to extend the above results to systems for positive and negative solutions. We prove our results by heavily utilizing a version of Lyapunov-Schmidt reduction method applied to a bifurcation from infinity.
We do not require any sign conditions on the nonlinear perturbations f and g near the origin. Thus Theorem 1.2 and Theorem 1.3 apply to both positone and semipositone type right-hand sides of (1.1) for positive solutions. See  and , where an asymptotically linear system is considered but nonlinearities are singular at the origin. The existence of a positive solution to the left of is established in  using Schauder fixed point theory and to the left of but away from is established in  using the sub and supersolution methods. In , , the authors consider a more general system, which includes system such as (1.1) as a special case. Their result, with additional assumptions on nonlinearities, shows that (1.1) has a positive solution in the right neighborhood of using critical point theory. These papers provide the existence of solutions, but do not provide information on the connectivity of the solution set. Therefore Theorem 1.2 and Theorem 1.3 complement these existence results.
As a by-product of the theorems above, we have the following existence results for (1.1) at and near resonance.
at least two solutions (one positive and one negative) for , and
at least one solution for .
at least one solution for , and
at least two solutions (one positive and one negative) for .
Theorem 1.4 and Theorem 1.5 generalize results for the scalar case in , Theorem 19] and , Theorem 9] to systems at the principal eigenvalue . Our results also complement , Theorem 2.1]. Also see – and , where existence results were discussed at and/or near resonance. We prove our results as a direct consequence of our main theorems. More precisely, Theorem 1.2 implies Theorem 1.4 and Theorem 1.3 implies Theorem 1.5.
Then we prove the following existence results, as corollaries of Theorem 1.4 and Theorem 1.5, respectively.
Then (1.9) has a solution.
Then (1.9) has a solution.
Similar results were obtained in , Theorem 5.1]. The author uses continuation of solution with respect to a parameter using the implicit function theorem and hence requires the autonomous part of the nonlinear perturbations to be smooth.
In Section 2, we set up the functional framework for our problem to apply the abstract bifurcation theory discussed in Appendix 1. In Section 3, a variant of Krasnosel’skii’s necessary condition for a bifurcation from infinity is discussed. In Section 4, we prove Theorems 1.1-1.3. In Section 5, we prove Theorems 1.4-1.5, and Corollaries 1.6-1.7. In Section 6, we provide several examples of f and g that satisfy the hypotheses of Theorems 1.1-1.3 and Corollaries 1.6-1.7. In Appendix 1, we discuss the abstract bifurcation theory (Rabinowitz and Dancer type) which we use in our analysis. In Appendix 2, we study the spectral properties of the eigenvalue problem (1.4). In Appendix 3, we provide the proof of (2.2) claimed in Section 2.
In this section, we set up functional framework that enables us to treat our problem using Proposition A.1 discussed in Appendix 1.
Due to the growth condition (H2) imposed on f and g and the fact that , all solutions of any elliptic system throughout this paper are understood in the strong sense, which means (bootstrap method and , Theorem 9.15]) and they satisfy the corresponding pde a.e. in Ω. Since, for any , for some , the boundary conditions are satisfied in the usual sense.
is well defined, obviously linear, continuous, and compact. Thus L is linear, continuous, and compact. Since f and g are Carathéodory functions satisfying (H2), the corresponding Nemytski operator, denoted again by f and g, maps continuously (see , Theorem 2.3]). Thus by the compactness of , we find that is continuous and compact.
uniformly for λ in compact intervals. See Appendix 3 for a proof.
is continuous and compact. Therefore the operators L, H, and 0H satisfy the hypotheses of Proposition A.1.
The following proposition is a variant of necessary condition for a bifurcation from infinity due to Krasnosel’skii , Statement, p.194]. This variant provides extra information on the convergence which is crucial in determining the definite sign (positive or negative) of solutions bifurcating from infinity.
where the convergence is in, for some.
with a ‘+’ sign, the other with a ‘−’ sign (one of the two subsequences may be void).
and hence satisfies (1.4). Since , must be an eigenvalue of (1.4) and a corresponding normalized eigenvector.
For the case , the statement follows from the fact that is simple. □
Proof of Theorem 1.1
and and are the eigenfunctions of the linear operator L and its adjoint , respectively, corresponding to the eigenvalue (see Appendix 2 for details).
Therefore, since on ∂ Ω, it follows from (4.1) that (<0) and (<0) for n sufficiently large. Thus there exist such that any with and satisfy (<0) and (<0) in Ω and for any with for large n. This proves the first part of Theorem 1.1.
By Proposition A.1, there exist two continua and emanating, respectively, in the direction of positive and negative multiple of in E. Then in view of the -regularity of solutions and Proposition 3.1, () is the continuum containing large positive (negative) solutions corresponding to (). □
Proof of Theorem 1.2
where , , , and as . Therefore and for large n.
for n sufficiently large.
and the last inequality follows from assumption (1.5). The inequality (iii) is a contradiction to (4.9). Therefore, for , for large n by (4.8). This means that there is no positive solution for for large n and thus part (I) assertion of Theorem 1.2 holds for positive solutions.
Proof of part (II) for positive solutions: Now it follows from Theorem 1.1 and part (I) above that the continuum , from Theorem 1.1, bifurcates from infinity at to the left. This concludes the proof of Theorem 1.2 for positive solutions.
and using instead of in applying Fatou’s lemma using (H4−) and reversing the inequalities appropriately. □
Proof of Theorem 1.3
The proof of Theorem 1.3 is similar with obvious changes. □
In , for the scalar case, the integral is analyzed for definite sign using as a parameter. For the case of systems, we use instead of the norm of , which allows us to analyze crucial integrals using only the parameter .
Proof of Theorem 1.4
By Theorem 1.2, it follows that for there exist a positive solution on the continua and a negative solution on the continua . This proves part (a).
which is absurd. Then by the Schauder fixed point theorem, (5.1) has a solution for each for any and hence for (1.1). In particular, (1.1) has at least one solution for all .
To complete the proof of part (b), it remains to show that (1.1) has a solution for . Theorem 1.1 implies that any solution of (1.1) with and are either , or , in Ω. But part (I) of Theorem 1.2 implies that there are no positive or negative solutions with and . Therefore all solutions with must satisfy the uniform bound .
that is, it satisfies (1.1) for . This establishes the existence of a solution in the resonant case. Thus the proof of Theorem 1.4 is complete. □
The proof of Theorem 1.5 follows similarly with appropriate changes.
Proof of Corollary 1.6
Therefore, by Theorem 1.4, the system (1.9) has a solution. □
The proof of Corollary 1.7 follows similarly with appropriate changes.
In this section we provide several examples of nonlinear perturbations f and g satisfying the hypotheses of our theorems. All examples below satisfy the hypotheses of Theorem 1.1.
(Unbounded perturbation: semipositone case)
Thus the hypotheses of Theorem 1.2 are satisfied for .
(Bounded perturbation: semipositone case)
Thus the hypotheses of Theorem 1.3 are satisfied for .
Let , .
Thus the hypotheses of Theorem 1.2 are satisfied.
Let and .
Thus the hypotheses of Theorem 1.2 are satisfied for .
(Landesman-Lazer type perturbation)
if then (1.9) has a solution provided satisfies (1.10), and
if then (1.9) has a solution provided satisfies (1.11).
We shall use results from the abstract bifurcation theory to prove our existence results. In particular, the theory of a bifurcation from infinity was developed mainly by Rabinowitz . It is well known that the result on a bifurcation from infinity  is developed from the theory of a bifurcation from zero . However, it was pointed out by Dancer  that the proofs of Theorem 1.27 and Theorem 1.40 in  contain gaps. Moreover, as pointed out in , the results from  for a bifurcation from zero are also wrong in their statement. Thus the statement and the proof of the result on a bifurcation from infinity  should be revisited in the spirit of discussion in –. We were not able to find these corrected statements and proofs for a bifurcation from infinity anywhere in the literature. Thus we restate and prove a corrected version of , Theorem 1.8] for the sake of the readers.
uniformly on compact λ-intervals.
It was shown by Krasnosel’skii , p.194] that a necessary condition for to be a bifurcation point from infinity is that μ is a characteristic eigenvalue of L.
Assume in addition that
possesses an unbounded component (continuum) which meets and satisfies the dichotomy of , Theorem 1.6]. In other words, meets , or meets where is another characteristic value of odd multiplicity, or is unbounded in the λ-direction.
We now state the correct result as intended in , Theorem 1.8] and provide the proof below.
Let L, H, and0H be as above. Let μ be a characteristic value of L of multiplicity 1, and letsatisfyingandbe fixed. Then there existand two continua, , of solutions to (A.1) such that for all: . Moreover, implieswith: , , andandas.
and denote the component of such that .
either and are both unbounded,
Now let us assume that and are both unbounded. Then they must leave the bounded sets through , and thus the intersections where are nonempty. If at least one of or is bounded, then, by the second part of the dichotomy, we have . This entails that both continua must leave through . This shows that in both cases of the dichotomy . It is worth noting that each component of the set has a nonempty intersection with the set but there may exist components that do not have in their closure. For each , there exists a component of , denoted by , such that and . Indeed, it suffices to show the existence of such a component satisfying . We prove by contradiction. Suppose there exists such that all components of are disjoint from by the ball , that is, . But this is not possible since and is a closed connected set.
Now let . Taking we see that and . Moreover, it follows from , Lemma 1.24] that implies with : , , and and as . Since , we find that for any . This completes the proof. □
With these eigenvalues being ordered, , let denote the eigenfunction corresponding to . By the standard regularity argument, , for any , and it is a strong solution of (1.2) with .
associated to problem (1.1). Note that the eigenvalues of (B.1) are the characteristic values of the linear operator L that appears in (2.1). We wish to point out that the eigenvalue problems are well studied in the literature for systems, including a more general case than (B.1). We state and prove the property that is necessary for our analysis. Interested readers will find the following references helpful for general linear eigenvalue problems:  for a system of n equations and , p.106] for a system of two equations; also see the references therein.
If we order these eigenvalues such thatforandfor, thenforandforare the eigenfunctions corresponding to, respectively. In particular, is the only eigenvalue value such that both components of its corresponding eigenfunction are positive in Ω.
Since the operator associated with the linear part of (1.1) is not self-adjoint, we will be dealing with the eigenfunction of (B.1) as well as of the corresponding adjoint equation.
It is well known that the eigenvalues of the adjoint equation corresponding to (B.1) are same as that of (B.1) but eigenfunctions are given by for and for .
which is in the original notation. Analogously, the eigenfunctions of (B.2) corresponding to are . Note that is the only eigenfunction of (1.2) which does not change sign in Ω and hence can be normalized to be positive on Ω. Thus is the only eigenvalue of (B.1) such that both components of its eigenfunction are positive in Ω. □
Proof of (2.2)
uniformly for . This establishes (C.1) and hence the proof of the claim is complete. □
The second author was supported by the Grant Agency of the Czech Republic Project No. 13-00863S.
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