Positive steady states of an indefinite equation with a nonlinear boundary condition: existence, multiplicity, stability and asymptotic profiles
We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter , which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from infinity occur at . Our approach combines variational and bifurcation techniques.
Key words and phrases:Semilinear elliptic problem, Indefinite weight, Variational methods, Bifurcation approach, Population dynamics
1991 Mathematics Subject Classification:35J20, 35J25, 35J61, 35B32, 35P30
1. Introduction and main results
Let be a bounded and regular domain of with . In this article we are concerned with the problem
is the usual Laplacian in
and if then
is the outward unit normal to .
Our main goal in this article is to carry on the study of , which was addressed in  for the logistic case . By variational and bifurcation techniques, we established existence and multiplicity results for non-negative solutions of . Moreover, the structure of the non-negative solutions set was also discussed. We intend now to deal with these issues in the case where changes sign.
By a solution of we mean a weak solution, i.e. satisfying
In this case, we may also say that the couple is a solution of . As already pointed out in , solutions of satisfy for some and , so that by the weak maximum principle [17, Theorem 9.1], nontrivial non-negative solutions of are strictly positive in .
describes the steady state of solutions of the corresponding initial boundary value problem
which appears as a model in population dynamics (see Cantrell and Cosner , Gómez-Reñasco and López-Gómez ). Here the unknown function stands for the population density of some species having as intrinsic growth rate and as extrinsic growth rate. If then the latter one is the well known logistic growth rate, with self-limitation () or without limitation (), so that the region where can be considered as a refuge. We can give the case the following biological interpretation (cf. ): in this case, the extrinsic growth rate measures the symbiosis effect due to the intraspecific cooperation whereas, in the case , it measures the crowding effect associated with competition. As for the nonlinear boundary condition, it suggests that the flux rate of the population on is incoming or outgoing (according to the sign of ) and depends nonlinearly on as (cf. ). From the population dynamics viewpoint, we point out that the parameter appearing in describes the reciprocal number of the diffusion coefficient , and only non-negative solutions are of interest.
Elliptic problems with indefinite nonlinearities have been studied over the last 25 years, starting with the works of Bandle, Pozio and Tesei , Ouyang , Alama and Tarantello , Berestycki, Capuzzo-Dolcetta and Nirenberg [5, 6], Lopez-Gomez , etc. Brown and Zhang  and Brown  used the Nehari manifold (or fibering) method to discuss existence, multiplicity, and non-existence of positive solutions for the problem
according to the position of . The sublinear case and the superlinear case were treated in  and , respectively. We shall see in this article that the Nehari manifold method turns out to be efficient for as well.
We note that is characterized by the combination of the nonlinearities in and on . Furthermore, the signs of , and may completely change the effect of these nonlinearities. For elliptic problems with such combined nonlinearities, we refer to Chipot, Fila, and Quittner , López-Gómez, Márquez, and Wolanski , Morales-Rodrigo and Suárez , Wu . Besides investigating existence and multiplicity of positive solutions, we shall analyze as well the structure of the positive solutions set. Our approach is mainly based on a detailed study of the energy functional associated to . Note however that unlike most of the aforementioned works, lacks coercivity on its left-hand side, since the term does not correspond to the norm of in .
Since the set of solutions of at is explicitly provided by the constants, we may obtain positive solutions for small by a bifurcation analysis on the line , where is a constant, cf. [28, 29]. On the other hand, the lack of continuity of the derivative of at prevents the use of the bifurcation approach to obtain positive solutions bifurcating from the zero solution. Finally, let us remark that the boundary point lemma cannot be applied directly to our problem, since it seems difficult to deduce that any nontrivial non-negative solution belongs to in view of the assumption . Therefore we are not able to infer that nontrivial non-negative solutions are positive on . However we shall prove in Proposition 5.1 that if is a nontrivial non-negative solution of then the set has no interior points in the relative topology of . Note that, in the one-dimensional case , the boundary point lemma is applicable, and we can deduce that any nontrivial non-negative weak solution of is positive on , see Proposition B.1 in Appendix B.
Let us set the notations and conventions used in this article:
The infimum of an empty set is assumed to be .
Unless otherwise stated, for any the integral is considered with respect to the Lebesgue measure, whereas for any the integral is considered with respect to the surface measure.
For the Lebesgue norm in will be denoted by and the usual norm of by .
We set and for .
The strong and weak convergence are denoted by and , respectively.
The positive and negative parts of a function are defined by .
If then we denote the closure of by and the interior of by .
The support of a measurable function is denoted by supp .
If is a functional defined on , we set
is a principal and simple eigenvalue of the problem
It is well known (cf. Brown and Lin ) that if and only if , in which case is achieved by a unique non-constant eigenfunction such that on . If then is achieved by .
Let be given by
It is easily seen that if either
then has two zeros , which satisfy
then has an unique zero, denoted by .
In order to simplify the notation, we introduce the functionals
defined on . From the compactness of the Sobolev embeddings
for and respectively, it is straightforward that is weakly lower semi-continuous, whereas and are weakly continuous.
The following results, which can be easily verified, will be used repeatedly throughout this article:
If is a bounded sequence in then we may assume that , , , and , for some .
If in and then in .
Let us set
is defined for . One may easily check that is quadratic, i.e. for and . Lastly, we set
We are now in position to state our main results. To begin with, we state an existence result, which provides bifurcation from zero and from infinity at in the following sense:
It is said that bifurcation from zero occurs at for if there exist nontrivial nonnegative solutions of such that , and in as . Similarly, it is said that bifurcation from infinity occurs at for if there exist nontrivial nonnegative solutions of such that , and as .
If then . If, in addition, then has a nontrivial non-negative solution for , which satisfies in for some as . Moreover, there exists such that in as , where is a nontrivial non-negative solution of
Furthermore, in , the set has no interior points in the relative topology of , and it is contained in if .
If then . If, in addition, then has a nontrivial non-negative solution for , which satisfies in for sufficiently small and as . Moreover, there exists such that in for some as , where , , is a solution of the problem
Furthermore on .
Theorem 1.3 states that if and then has, besides the trivial branch , the solution bifurcating from zero at . In a similar way, if and then has, besides the trivial branch of positive constants, the solution bifurcating from infinity at . Furthermore, the blow up of as occurs uniformly on .
We shall see in Proposition 5.1 that for any such that there holds if is sufficiently small. More precisely, if is such that then for sufficiently small.
Our next results provide multiplicity of non-negative solutions and are stated under the condition
Since we are considering , the case reduces to (1.13) after the change of variable .
Assume (1.13), , and . Then:
has a nontrivial non-negative solution for . Moreover in for some as .
has two nontrivial non-negative solutions , for , where and . Furthermore, and in for some as .
Under the assumptions of Theorem 1.5 and stronger regularity conditions on , we shall obtain, for sufficiently small, a positive solution of converging to as . This will be carried out by a bifurcation argument and, as a consequence, will provide at least four nontrivial non-negative solutions of for small enough. Since this argument requires only the existence of zeros of and some regularity on , and , we shall assume, in addition to (1.13),
It is easily seen that , so that .
Hereafter, by a classical positive solution of we mean for some which satisfies in the classical sense and is strictly positive on .
Since , solutions of (1.12) are in with , so that the boundary point lemma is not applicable, and consequently we can not deduce the positivity of on . If this is the case then we can prove a result similar to Theorem 1.6 for , obtained in Theorem 1.3. More precisely, let us assume (1.14) and on . If changes sign and then, for sufficiently small, is a classical positive solution of which is moreover asymptotically stable. We include a sketch of the proof of this result in Section 5.
is a classical positive solution of which is asymptotically stable for . Moreover, has a classical positive solution for which is unstable. These solutions are continuous in with respect to and emanate from and respectively, i.e. and . Finally, there are no other classical positive solutions of converging to some positive constant in as .
are classical positive solutions of for , which are asymptotically stable, and unstable, respectively. Moreover, these solutions are continuous in with respect to and emanate from and respectively, i.e. and . Finally, there are no other classical positive solutions of converging to some positive constant in as .
Under the assumptions of Theorem 1.5, assume moreover that and . Then has three nontrivial non-negative solutions , , for , which satisfy and in for some and as .
We shall see in Proposition 5.3 that under (1.13), (1.14), and (1.15), has, for sufficiently small, two classical positive solutions , , which are continuous (with respect to ) in and satisfy , . Furthermore, this result does not require the condition . So, under (1.13), (1.14) and the conditions
If then has at least four nontrivial non-negative solutions (two variational nontrivial non-negative solutions, among which one is a classical positive solution, and two bifurcating classical positive solutions) for sufficiently small.
If then has at least two (variational) nontrivial non-negative solutions, among which one is a classical positive solution, for sufficiently small. Moreover, if then has no classical positive solutions converging to a positive constant as .
The following condition provides an a priori bound on the values of for which has a nontrivial non-negative solution:
Assume that changes sign and holds. Then there exists a constant such that if is a nontrivial non-negative solution of then .
As a particular case of , we shall consider the problem
is defined for such that . Note that and with .
The assumption of Theorem 1.5 reads now
in which case has two positive zeros .
Assume (1.17) and changes sign. Then and:
has two nontrivial non-negative solutions , for , which satisfy in for some and as . If, in addition, , then has a further nontrivial non-negative solution for , which satisfies in as .
has two nontrivial non-negative solutions , for , where and . Furthermore, and in for some as .
Lastly, we consider the case where has a unique positive zero such that , which occurs precisely when
The following result asserts that if (1.18) holds then is a turning point to the right on the smooth curve of positive solutions of .
Assume (1.14). If has a classical positive solution with such that in as , where is a positive constant, then . Moreover:
Assume (1.15) with . Then there exist two arbitrarily smooth maps for close to such that , are classical positive solutions of and satisfy and .
Assume (1.18). Then there exist two arbitrarily smooth maps and for close to such that is a classical positive solution of with , , and as . Moreover, there exists a constant such that is asymptotically stable for and unstable for .
The results on the number of nontrivial non-negative solutions from Remark 1.12 can be sharpened for . In this case, Theorems 1.3 and 1.15 provide that, under (1.14), if and change sign, and , then has:
at least four nontrivial non-negative solutions (two variational nontrivial non-negative solutions, among which one is a classical positive solution, and two bifurcating classical positive solutions) for sufficiently small if .
at least two (variational) nontrivial non-negative solutions, among which one is a classical positive solution, for sufficiently small and no classical positive solutions converging to a positive constant as if .
1.1. Suggested bifurcation diagrams
In view of the results stated above, we analyze now the possible bifurcations diagrams of and . This will be done assuming that changes sign and holds, in which case Theorem 1.13 ensures that has no nontrivial non-negative solution for .
Assume (1.13), and change sign, and . If then, by Theorems 1.3, 1.5, and 1.8, the bifurcation diagram for is suggested by Figure 1. Moreover, Remark 1.12 suggests that this bifurcation diagram approaches the one shown in Figure 2 as . Indeed, note that the value provided by Theorem 1.13 does not depend on . Furthermore, it can be shown that and stay bounded away from zero if is bounded from above (cf. Remark 3.5). By a formal observation, the nonlinear boundary condition in approaches the Dirichlet boundary condition as . So, after the change of variable for , the limiting problem for when would be
This problem has been investigated by Ouyang  in the case and the existence of a single turning point in the positive solutions set has been proved under some conditions on . We expect then that the bifurcation diagram of has a unique turning point if .
Assume (1.13), , changes sign, and .
If then the bifurcation diagram for in the case is suggested by Figure 3. This is motivated by Theorems 1.3, 1.5, and 1.8, and the fact that approaches zero as converges to zero. More precisely, in Proposition 3.6 we prove that if in with for every , then there exists such that the solution exists for every , and satisfies in for some and . Note that in  we have obtained bifurcation from zero in the case and when .
From the biological viewpoint, the bifurcation diagram suggested in Figure 1 would provide three possible conditional states for the population as the diffusion rate grows to infinity, namely: extinction, explosion and persistence with a spatially uniform distribution.