This work is devoted to the study of existence and multiplicity of vector solutions of the problem
with the boundary condition
Here L denotes the vector Laplacian given by , is a smooth bounded domain and is a smooth function. Without loss of generality, it shall be assumed throughout the paper that . We shall also assume that and . It is remarked that, unlike in the most standard Robin condition, we do not assume that .
The problem arises in many different applications; for instance (see  and the references therein) for some particular , it can be seen as the steady state of the population density of some species, governed by a parabolic problem. The boundary condition corresponds to the law of the population flux, given on the boundary by γ. Thus, the inflow/outflow of population to the region occurs when or , respectively. Our direct motivation for the present work can be found in , where a Painlevé II (ordinary) scalar equation was considered, namely
under the radiation boundary conditions , with . It was shown that is compensated by the effect of the superlinear term in such a way that the associated functional is still coercive, giving rise to a unique solution when and exactly three solutions when , provided that the positive parameter p is small. It is worth noticing that, in this model, δ is a linear function depending on and ; thus, the uniqueness/multiplicity of solutions can be understood as a consequence of the interaction between the nonlinear term of the equation and the Robin coefficients. The previous results have been extended in  and  for the equation , where is a general superlinear function, that is,
uniformly on x. Indeed, it was shown that the interaction between and can be expressed in a very precise way in terms of the spectrum of the associated linear operator under the Robin conditions: if
for all x and all u, then the problem has a unique solution, and if
for some odd k, then the problem has at least three solutions when is small. Furthermore, if (1.4) holds with k even, then the problem has at least five solutions for small p, provided that .
The main purpose of this paper consists in obtaining suitable extensions of the results mentioned above in two directions: in the first place, we shall consider a system instead of a scalar equation and in the second place, we shall not be longer dealing with ODEs but with a PDE system. We shall study system (1.1) in terms of the eigenvalues of the scalar problem associated to the Robin condition and . At first sight, it seems reasonable to assume that is superlinear, that is,
uniformly on x, where and denote the usual inner product and norm of , respectively. However, we shall employ a weaker condition, namely, that there exists a constant depending only on Ω and γ such that, if
uniformly on x, then the problem has a solution (see Theorem 3.3). This constant can be estimated: for example, it will be shown that it always suffices to take
where is the (unique) principal eigenfunction such that . For the ODE case or when g grows like for some , it can be seen that the condition is optimal for our purposes.
Extra solutions will be obtained when p is small, under an appropriate condition on the interaction between and the spectrum of the linear associated operator. In more precise terms, if for simplicity we assume that for some smooth function , then our main hypothesis is that the Hessian matrix at satisfies
where are symmetric matrices with eigenvalues and such that
for some and all . Here, the ordering over the set of symmetric functions is understood in the usual sense that if and only if is positive semi-definite or equivalently, for all .
It shall be seen that if is an odd number, then the problem has at least two solutions (and generically three), provided that is small enough. In fact, the result holds for a general function , with the role of the Hessian matrix now played by the Jacobian matrix , provided it is symmetric. In contrast with the case , the general situation does not have variational structure and is presented in our main result through the Leray–Schauder degree method (see Theorem 5.2 below). To our knowledge, there are no previous similar results in the literature concerning the multiplicity of solutions for an elliptic system with indefinite Robin conditions.
The paper is organised as follows. In the next section, we recall some basic facts concerning the spectrum of the associated linear operator. In Section 3, we prove the existence of at least one solution by means of an appropriate space-dependent Hartman condition. In Section 4, we obtain bounds for all solutions, provided that . Furthermore, enlarging μ if necessary, we also obtain bounds, which shall be the key for proving our multiplicity result. This task is accomplished in Section 5, where we firstly prove a lemma that helps to compute the degree of certain linear operators and then apply the lemma for the computation of the degree of , where K is an appropriate compact fixed point operator. Roughly speaking, for we shall prove that the degree is equal to 1 over a large ball and equal to -1 over a small neighbourhood of the origin. This guarantees, when p is small, the existence of a solution close to the origin and typically two ‘large’ solutions.
For convenience, throughout the paper we shall employ the notation . It is clear that if satisfies (1.5), then so does g for arbitrary . Sometimes we shall express the condition in the following equivalent form: there exist and such that
Before establishing our main results, we need to recall some basic facts concerning the spectrum of the associated linear operator. Let us observe, in the first place, that the scalar operator is symmetric for the Robin boundary condition. Thus, by standard arguments (see e.g. ) one can deduce the existence of a sequence of eigenvalues and an associated orthonormal basis of consisting of eigenfunctions . For convenience, we shall denote .
It is verified (see ) that is simple and the associated eigenfunctions do not vanish on . From now on, we shall always assume that is such that . Moreover, can be obtained as a global minimiser of the functional
subject to the restriction . It follows that
for any smooth function φ satisfying the Robin condition. In particular, adding to both sides for some , the Cauchy–Schwarz inequality yields the following estimate:
Using the standard norm of still denoted by , namely
we observe that (2.2) and (2.3) can be extended in an obvious way for vector functions, replacing Δ by L. In particular, the latter inequality implies, as an immediate consequence, the uniqueness of solutions, provided that is (strictly) monotone increasing, that is,
for all x and . Clearly, this extends condition (1.3) for .
In order to understand the dependence of with respect to γ, observe that if is the functional associated to some and if are the respective eigenvalue and eigenfunction, then, by virtue of the trace embedding , we deduce that
We conclude that if uniformly, then . Furthermore, since we deduce that the family is uniformly bounded in , and by a bootstrapping argument, we conclude that it is uniformly bounded in . We claim that for the norm. Indeed, otherwise we may suppose that converges for the norm to some . Observe that and ; moreover, since , it is readily seen that and hence , a contradiction.
3 A space-dependent Hartman condition
Our general existence result is based on the fact that g satisfies the following Hartman type condition (the original condition was formulated in ):
There exists a smooth function such that
for all and all such that .
Assume that there exists as before such that (3.1) holds and
Let . For fixed , a straightforward application of Schauder’s theorem shows that the truncated problem
has at least one solution u. We claim that, in view of (3.1) and (3.2) it can be deduced that for all x and, consequently, u is a solution of the original problem (1.1)–(1.2). Indeed, otherwise the function achieves a strictly positive maximum value at some point . If , then
for all j and , that is,
where we employed the notation . Then
Moreover, observe that
and consequently, using (3.1), we obtain
Now suppose , then
This implies that
which contradicts (3.2). ∎
In order to establish the existence of solutions, let us observe that, without loss of generality, we may assume that for some mapping such that 0 is a regular value of F. Thus, the outer normal at is simply computed as . For convenience, let us fix the following constants:
Define with a as before and some to be specified. Direct computation shows that
for all . On the other hand,
with C given by (3.3). Then we may fix such that
for and such that for all x. Hence, for it is seen that
and the previous lemma applies. ∎
4 A priori bounds for the solutions
To this end, let us firstly observe that necessarily : indeed, for we may consider
for which the problem has an unbounded set of solutions if all the coordinates of p are orthogonal (in the sense) to the eigenfunction associated to , and no solutions otherwise.
We claim that the condition is also sufficient for getting appropriate bounds. Indeed, using the extension of (2.2) for vector functions (see Remark 2.1), we can multiply the equation by u and integrate to obtain
From inequality (2.1) and using the fact that it is verified that , which in turn yields
Hence and, moreover,
Next, extend the outer normal to a smooth vector field and define . Then
Now let and assume, without loss of generality, that . Since
it follows that
where and the desired bound is obtained.
Thus we have proved the following lemma.
The preceding argument shows that, if , then the associated functional is coercive. However, the functional is not defined over unless a growth condition is assumed for G. As we shall see, by enlarging μ in an appropriate way, we can still apply the variational method because g may be replaced by a suitable truncation.
With this aim, assume that (2.1) holds with and set as before such that (3.2) holds strictly. Fix as in the proof of Theorem 3.3 and set b large enough such that for all x. From the previous computations it is deduced that if u is a solution and for some x, then the absolute maximum of the function is achieved at the boundary. Indeed, with the previous notation observe that if the (nonnegative) maximum value of ϕ is achieved at some , then it is verified as before that
a contradiction. Let u be a solution and suppose that for some x. The function is continuous with respect to b, so (since as ) enlarging b, if necessary we may assume that , that is, for and for some . This implies that
In summary, there exists a constant b such that if u is a solution of the problem, then . Furthermore, observe that the choice of R depends only on the fact that for and the constant C itself is independent of g; thus, replacing g by the function
we see that the resulting problem has the same solutions. Hence, we may assume that g grows linearly with respect to u. In particular, if , then the existence of at least one solution is also deduced by variational methods (see Remark 4.3).
An alternative choice of R in the Hartman condition is
for some c sufficiently large, provided that
for , that is,
Thus, it suffices to consider
Obviously, this R does not satisfy (3.2) strictly, so the previous argument for the bound fails. However, we may overcome this difficulty by taking, instead, for some η sufficiently small such that with ε as in (2.1) and , which satisfies the Hartman condition and
5.1 A useful lemma
The key for a proof of our multiplicity result shall be the computation of the degree of an associated linear operator. For and a fixed continuous matrix function let us define the compact linear operator given by , the unique solution of the problem
Then we have the following lemma.
Let be a symmetric matrix function such that , where the symmetric matrices satisfy (1.6). Then . Furthermore, if is a bounded open neighbourhood of 0, then
Following the ideas in , we can verify that the problem
has no nontrivial solutions satisfying the boundary condition (1.2). This implies that vanishes only at and, consequently, is well defined.
Next, observe that can be computed as
where is the Green function associated to the scalar operator under the Robin boundary conditions and
with the identity matrix . Recall that can be written as
where is an orthonormal basis of eigenfunctions associated to the eigenvalues . It is clear that and we may take . Thus,
Moreover, it is seen that the degree, regarded as a function of M, is continuous, that is, locally constant. Hence, using the fact that the subset of defined by is connected (because it is convex), we may assume that M is a constant matrix, say .
Writing , the coordinate of the vector function u in the basis , we obtain
Observe also that
uniformly for . This implies that, for some large enough q, it suffices to compute the degree restricted to the subspace spanned by or, equivalently, the degree of the mapping
In other words, it suffices to compute the sign of the determinant of the block matrix
Let be the eigenvector of the matrix associated to the eigenvalue , then
and, since , it follows that
Thus, if for all k, it is seen that
and the result is deduced from the fact that
5.2 Multiple solutions for p small
for some , and all such that
for all x. If is odd, then there exists such that the problem has at least two solutions for all p such that .
Assume firstly that . According to Section 4.1, we may fix a constant depending only on μ and such that for any solution u. Moreover, as mentioned in Section 4.2, we may also assume that has linear growth; thus, for fixed , the operator given by , the unique solution of the problem
is well defined and clearly compact. Let be defined as in Lemma 5.1, with , then the operator has no fixed points on for , because satisfies (2.1). Since , the assumptions of Lemma 5.1 hold with and for all k, then
Next we shall prove that if is small enough then on and
To this end, set and observe that if , then by letting , we deduce from estimate (2.3) that
Due to the linearity and compactness of , it is easy to verify that there exists a constant such that
for all . It follows, for sufficiently small, that the operator defined by has no fixed points on for , because
for . This implies, for small ρ, that the degree of is well defined and, according to Lemma 5.1,
which proves the claim. Moreover, by the excision property of the degree,
Since the degree is locally constant with respect to the third coordinate, it is deduced that, for any sufficiently close to 0, the degrees and are well defined and equal to -1 and 2, respectively.
In order to complete the proof, for each we define as the unique solution of the linear problem
The previous estimate (2.3) yields ; that is, if p is close to 0 then so is P and the problem has at least one solution in and another one in . The result is thus deduced from the fact that if , then
Using the Sard–Smale theorem (see ), we deduce that the problem has generically at least three solutions, namely, there exists a residual set such that if , then the problem has at least three solutions.
6 Further discussion and open problems
The existence of solutions for problem (1.1)–(1.2) was obtained under condition (1.5). Remarkably, the bounds of Section 4.1 imply by themselves that the degree of the operator defined in the proof of Theorem 5.2 over a large ball is equal to 1, thus proving the existence of at least one solution. Since the bounds require only that , one might be tempted to believe that the latter condition is sufficient for all our purposes; however, the operator K is not well defined for arbitrary g and this is why we needed also the bounds. As an immediate consequence, we conclude that the condition is optimal when . This is still true for , provided that g has polynomial growth and also when , provided that g behaves asymptotically as with . So it is an open problem to determine, for the general case, the optimal value of μ when . The value given by (4.1) is clearly larger than and also satisfies the following lower bound: let
Since , it is deduced that
On the other hand, observe that the use of a variable R in the Hartman condition can be avoided if and . The fact that is compensated by enlarging the value of μ. It is worth mentioning that, as shown in , the problem can always be reformulated into an equivalent one with ; however, under this transformation the relation between and the eigenvalues becomes less clear.
Also, it would be interesting to analyse the case in which is not necessarily symmetric. It is worth noticing that symmetry was used only in order to apply the lemma on bilinear symmetric forms used in , but, at first sight, more general results are possible. This is left as a second open problem.
Another matter concerns further multiplicity. In the same line of Theorem 5.2, one may try to see, when is even, if it is possible to obtain five solutions, as it happens for the ODE case presented in . This is true for the scalar case , for which the function defined in Section 3 can be used as an upper solution and, if in , then with small enough serves as a lower solution. Taking
we have that
In the same way, an ordered couple of a lower and an upper solution is readily obtained. Furthermore, we may take small enough such that
thus, if condition (1.4) is satisfied with even, then
so generically the problem has at least five solutions. Moreover, observe that the condition is fulfilled automatically if k is sufficiently large: for instance, for the ODE studied in  with , it suffices to take because it is verified that .
Finally, we remark that condition (1.6) makes sense only when . Unlike the case , for which an elementary study of the Wronskian determinant shows that all the eigenvalues are simple, nothing is known about the multiplicity of higher order eigenvalues when . This might reduce the number of possible different situations in Theorem 5.2: for example, for the periodic ODE one has
In particular, all the eigenvalues, except the first one, have even multiplicity. This means that, if (1.6) holds, then or is odd. Thus, the assumption that is odd simply says, in this case, that is odd.
The author thanks Professor Mónica Clapp for her thoughtful comments and to the anonymous referee for the very careful reading of the manuscript and fruitful corrections and remarks.
P. Amster and M. P. Kuna, Multiple solutions for a second order equation with radiation boundary conditions, Electron. J. Qual. Theory Differ. Equ. (2017), 10.14232/ejqtde.2017.1.37. Web of ScienceGoogle Scholar
About the article
Published Online: 2017-07-08
Funding Source: Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires
Award identifier / Grant number: 20020120100029BA
Funding Source: Consejo Nacional de Investigaciones Científicas y Técnicas
Award identifier / Grant number: PIP11220130100006CO
This work was partially supported by projects UBACyT 20020120100029BA and CONICET PIP 11220130100006CO.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 603–614, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0034.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0