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Asymptotically linear systems near and at resonance
Boundary Value Problems volume 2014, Article number: 242 (2014)
Abstract
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.
MSC: 35J60.
1 Introduction
We consider an elliptic system of the form
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;
(H2) there exist , with for a.e. , and a constant such that
(H3) f and g are sublinear at infinity, that is, for a.e.
Therefore, the system (1.1) is asymptotically linear at infinity.
In this paper, we are interested in studying positive solutions of (1.1) via bifurcation theory. For this, we define the underlying space
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).
We say that is a bifurcation point from infinity if the solution set
contains a sequence such that
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 .
Let be the principal eigenvalue of
and be the corresponding eigenfunction. Without loss of generality, we normalize the eigenfunction such that in Ω and
Define . We will show, in Appendix 2, that is a simple eigenvalue of the linear operator associated with the linear part of (1.1), that is, of
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.
Theorem 1.1
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.
For , define
and
Suppose there exist and nonnegative functions for such that
Now let ε, K, , be as in Theorem 1.1. Then we prove the following results.
Theorem 1.2
Suppose the hypotheses of Theorem 1.1hold. If (H4+) ((H4−)) holds and
then
-
(I)
there is no positive (negative) solution satisfying and , and
-
(II)
the continuum , consisting of positive (negative) solutions, bifurcates from infinity at to the left.
Theorem 1.3
Suppose the hypotheses of Theorem 1.1hold. If (H5+) ((H5−)) holds and
then
-
(I)
there is no positive (negative) solution satisfying and , and
-
(II)
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 [1], Theorem 3 and Theorem 4] and in [2], 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 [3] and [4], 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 [3] using Schauder fixed point theory and to the left of but away from is established in [4] using the sub and supersolution methods. In [5], [6], 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.
Theorem 1.4
Suppose the hypotheses of Theorem 1.1hold. If (H4+) and (H4−) hold, and
then (1.1) has
-
(a)
at least two solutions (one positive and one negative) for , and
-
(b)
at least one solution for .
Theorem 1.5
Suppose the hypotheses of Theorem 1.1hold. If (H5+) and (H5−) hold, and
then (1.1) has
-
(a)
at least one solution for , and
-
(b)
at least two solutions (one positive and one negative) for .
Theorem 1.4 and Theorem 1.5 generalize results for the scalar case in [1], Theorem 19] and [7], Theorem 9] to systems at the principal eigenvalue . Our results also complement [8], Theorem 2.1]. Also see [9]–[11] and [12], 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.
Finally, we consider a straightforward extension of Landesman-Lazer type conditions initiated in the celebrated paper [13] by Landesman and Lazer to a system. In particular, we provide the solvability of the following system at resonance,
We assume , , as before. Let be continuous functions and with satisfying
Then we prove the following existence results, as corollaries of Theorem 1.4 and Theorem 1.5, respectively.
Corollary 1.6
Suppose
Then (1.9) has a solution.
Corollary 1.7
Suppose
Then (1.9) has a solution.
Similar results were obtained in [10], 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.
Remark 1.8
In Figure 1, any solution in both dark and light gray regions will either be positive or negative by Theorem 1.1. There is no positive or negative solution in the dark gray regions due to Theorem 1.2. Solutions in the light gray region on are positive and on are negative. Dashed points are a sequence of solutions approaching the solution of the resonant problem at from the right (see proof of Theorem 1.4). The vertical axis is given by
in order to distinguish positive and negative solutions in the neighborhood of . Here the symbol ‘⋅’ denotes the usual scalar product in . To understand the diagram it is helpful to observe that for solutions with λ close to . See (4.4) for a precise statement. Values of corresponding to the value K from Theorem 1.1 are denoted by ‘K’ and ‘−K’.
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.
2 Functional framework
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 [14], 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.
The abstract setting of our problem is
where denotes the mapping
and denotes the mapping
The solution operator , , associated to the problem
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 [15], Theorem 2.3]). Thus by the compactness of , we find that is continuous and compact.
Claim: H satisfies
uniformly for λ in compact intervals. See Appendix 3 for a proof.
In order to satisfy the assumptions of Proposition A.1, we use the Kelvin transform
and define by
Clearly is continuous for on compact intervals of λ. The continuity at follows from the fact that
as and hence as . Similarly for g. Due to the compactness of it follows that defined as
is continuous and compact. Therefore the operators L, H, and 0H satisfy the hypotheses of Proposition A.1.
3 Bifurcation from eigenvalue
The following proposition is a variant of necessary condition for a bifurcation from infinity due to Krasnosel’skii [16], 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.
Proposition 3.1
Ifis a bifurcation point from infinity for (1.1), thenfor some. Moreover, for any sequencewithandas, there exists an eigenfunctionsatisfyingcorresponding to the eigenvalue, and a subsequenceofsuch that
where the convergence is in, for some.
In particular, for, any sequencewithandascan be split into two subsequences one of which satisfies
with a ‘+’ sign, the other with a ‘−’ sign (one of the two subsequences may be void).
Proof
Let be solutions of (1.1) such that and , where is a compact interval. Then satisfies
Owing to (C.1), we find that the terms on the right-hand sides are bounded in (independent of n). Hence and are bounded (independent of n) and so are and , for some . Since compactly for , passing to a subsequence, , in and . Therefore satisfies
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. □
4 Proof of main results
Proof of Theorem 1.1
We will use the Lyapunov-Schmitt reduction method combined with a bifurcation from infinity. We split our underlying space , where
and and are the eigenfunctions of the linear operator L and its adjoint , respectively, corresponding to the eigenvalue (see Appendix 2 for details).
By Proposition 3.1, for large n, takes the form
where , , and , that is, they satisfy the ‘orthogonality’ condition
with , in as . It follows from a straightforward calculation, using (1.3), that is the projection of to the subspace spanned by . Indeed,
It is important to observe that if and only if as . Indeed, it follows from (4.3), (1.3) and (3.2) that
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
We will first establish part (I) of Theorem 1.2 by determining the λ-direction of the bifurcation of positive solutions from infinity at . Let be such that and . Then by Proposition 3.1, takes the form
where , , , and as . Therefore and for large n.
Multiplying the first equation of (1.1) by the first component of (eigenfunction corresponding to the adjoint operator ), integrating over Ω, and using (4.1), we obtain
This yields
and, using (1.3), it simplifies to
Similarly, multiplying the second equation of (1.1) by the second component of , and integrating over Ω, we obtain
Adding (4.6) and (4.7), we have
Using and (4.2), the above equation simplifies to
The previous equation gives rise to the following important identity for sufficiently large n:
Proof of part (I) for positive solutions: Now we proceed to complete the proof of part (I) of Theorem 1.2 by contradiction. Suppose that for all n sufficiently large. This implies, by (4.8), that
for n sufficiently large.
On the other hand, since and for large n, we have
Then since as and , the following pointwise estimates hold in Ω as :
Consequently, using (H4+), we get the following pointwise estimate a.e. in Ω for sufficiently large n:
Similarly, for sufficiently large n
Observe that there exist , and , such that
Therefore, since and , the Hölder inequality asserts (see [1] for a detailed discussion). The estimates (4.10) and (4.11) allow for the use of Fatou’s lemma below yielding
The inequality (i) follows from Fatou’s lemma, (ii) follows since
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.
Proof of Theorem 1.2for negative solutions: The proof for negative solutions can be carried out in a similar fashion by using , defining
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. □
Remark 4.1
In [1], 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 .
5 Proofs of Theorems 1.4-1.5 and Corollaries 1.6-1.7
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).
For part (b), first we establish the result for the non-resonant case. For this, define to be the solution operator of
For , by the Fredholm alternative, the above system has a unique strong solution for any . Therefore and is continuous. Then it follows from the standard compactness argument that is compact. Next, define by . Then the operator equation corresponding to (1.1) is
Now we show that for each fixed , for any , there exists a constant such that . Indeed, suppose to the contrary that as . Since f and g satisfy (H3), it follows from (C.1) that
Dividing (5.1) by and taking the E norm yields
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 .
Now let be a sequence of solutions of (1.1) with such that . Then for each , satisfy
Since , independent of n, there exists a subsequence, denoted again by , that converges to some in for some . Then and in (cf.[15], Theorem 2.3]). Thus the right-hand sides of the above system converges in . By the same argument as in the proof of Proposition 3.1, in E and the limit satisfies
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
We will use Theorem 1.4 for with and . Then it is easy to see that hypothesis (1.7) is satisfied by (1.10) with
Therefore, by Theorem 1.4, the system (1.9) has a solution. □
The proof of Corollary 1.7 follows similarly with appropriate changes.
6 Examples
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.
Example 6.1
(Unbounded perturbation: semipositone case)
Let .
Then and satisfy
Thus the hypotheses of Theorem 1.2 are satisfied for .
Example 6.2
(Bounded perturbation: semipositone case)
Let .
Then and satisfy
Thus the hypotheses of Theorem 1.3 are satisfied for .
Example 6.3
(Vanishing perturbation)
Let , .
Then and satisfy
Thus the hypotheses of Theorem 1.2 are satisfied.
Example 6.4
(Oscillating perturbation)
Let and .
Then and satisfy
Thus the hypotheses of Theorem 1.2 are satisfied for .
Example 6.5
(Landesman-Lazer type perturbation)
Let , , and . Then
-
(i)
if then (1.9) has a solution provided satisfies (1.10), and
-
(ii)
if then (1.9) has a solution provided satisfies (1.11).
Appendix 1: Abstract bifurcation
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 [17]. It is well known that the result on a bifurcation from infinity [17] is developed from the theory of a bifurcation from zero [18]. However, it was pointed out by Dancer [19] that the proofs of Theorem 1.27 and Theorem 1.40 in [18] contain gaps. Moreover, as pointed out in [20], the results from [18] 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 [17] should be revisited in the spirit of discussion in [19]–[21]. 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 [17], Theorem 1.8] for the sake of the readers.
In their abstract setting, the bifurcations are studied in the product space where ℬ is a real Banach space with norm . For , we consider the norm . In what follows, by connected components of a topological space we mean the maximal connected subsets (ordered by inclusion) of the given space. We use the abstract bifurcation theorems concerning the bifurcation of continua of solutions from infinity for the operator equation
Here is the bifurcation parameter and is a linear and compact operator, is a continuous and compact operator which satisfies sublinearity condition at infinity, that is,
uniformly on compact λ-intervals.
It was shown by Krasnosel’skii [16], p.194] that a necessary condition for to be a bifurcation point from infinity is that μ is a characteristic eigenvalue of L.
Define
Assume in addition that
is compact,
as continuity follows from (A.2). Let be a characteristic value of L of odd multiplicity. It was proved in [17], Theorem 1.6] that the set
possesses an unbounded component (continuum) which meets and satisfies the dichotomy of [17], Theorem 1.6]. In other words, meets , or meets where is another characteristic value of odd multiplicity, or is unbounded in the λ-direction.
This abstract result for a bifurcation from infinity can easily be strengthened by using [19], Theorem 2], which applies to a bifurcation from zero from a characteristic value of multiplicity 1. For , define
We now state the correct result as intended in [17], Theorem 1.8] and provide the proof below.
Proposition A.1
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.
Proof
Indeed, bifurcations from infinity can be studied via bifurcations from zero using the Kelvin transform for , which turns solutions large in the norm to solutions small in the norm and vice versa. For we set . Note that for , reciprocally, and for . Now dividing (A.1) by we obtain . Since 0H is continuous and compact, the equation
satisfies the assumptions of [19], Theorem 2], which we describe briefly below for clarity and completeness of the proof. For this, we borrow the notation from [19]. Let be a characteristic value of L of multiplicity 1, , and satisfy and (where is adjoint to L) and . For , define
In particular, , , are convex cones, , and , where −ν stands for the sign opposite to ν. Let denote the closure of all nontrivial solutions of (A.3) in , that is,
and denote the component of such that .
For any , by [18], Lemma 1.24], there exists such that
Then for define
and for define to be the component of containing . Let be the component of containing (here again for ). Finally, we define to be the closure of in . Then by [19], Theorem 2] the following dichotomy holds:
-
(i)
either and are both unbounded,
-
(ii)
or .
Now fix . Then for all , one has
by (A.4). By the respective definitions of where , we see that for all ,
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 [18], Lemma 1.24] that implies with : , , and and as . Since , we find that for any . This completes the proof. □
Appendix 2: Eigenvalue problem
Let , with denote the j th eigenvalue of
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 .
Now we describe the eigenvalues and eigenfunctions of the linear system
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: [22] for a system of n equations and [23], p.106] for a system of two equations; also see the references therein.
Proposition B.1
The eigenvalues of (B.1) form the following set:
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.
Remark B.2
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 .
Proof
Using matrix notation for convenience, the eigenvalue problem (B.1) reads as follows:
Note that and ,
are respective eigenvalues and eigenvectors of the coefficient matrix. Taking into account the linearity of ‘−Δ’, we infer that
satisfies
The equations of this system are not coupled and it is obvious that if and only if and for some . On the other hand, if and only if and for some . Therefore, imply and imply . Hence, the eigenfunctions of (B.2) corresponding to are
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. □
Appendix 3: Proof of (2.2)
Proof of (2.2)
It is enough to show that
as uniformly for λ in compact intervals for any sequence such that . We will show the convergence of the first sequence and the proof of the second is identical. Let be a compact interval. It follows from (H2) that
As in [24], for each , we define sets
Both sets are measurable and for any . For a.e. , we have
On the other hand, by the definition of
for a.e. . Since as , there exists such that for all . It follows from (C.6) that
for all . Applying (C.2), (C.6), and (C.7) in (C.5) yields
a.e. in and for all . For a.e. , we have
where C is the constant of the embedding . Then, for ,
Now let and denote the characteristic functions of and , respectively. Then, for ,
for a.e. since by (H3). Hence by the Lebesgue dominated convergence theorem (Ω is bounded, integrand dominated by 1) one can assert that
uniformly for . This establishes (C.1) and hence the proof of the claim is complete. □
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The second author was supported by the Grant Agency of the Czech Republic Project No. 13-00863S.
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Chhetri, M., Girg, P. Asymptotically linear systems near and at resonance. Bound Value Probl 2014, 242 (2014). https://doi.org/10.1186/s13661-014-0242-z
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DOI: https://doi.org/10.1186/s13661-014-0242-z