An indefinite concave-convex equation under a Neumann boundary condition I
We investigate the problem
where is a bounded smooth domain in (), , , and with . Under some indefinite type conditions on and we prove the existence of two nontrivial non-negative solutions for small. We characterize then the asymptotic profiles of these solutions as , which implies in some cases the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type subcontinuum in the non-negative solutions set. We prove in some cases the existence of such subcontinuum via a bifurcation and topological analysis of a regularized version of .
Key words and phrases:Semilinear elliptic problem, Concave-convex nonlinearity, Indefinite problem, Non-negative solution, Bifurcation, Variational methods, Loop type subcontinuum
1991 Mathematics Subject Classification:35J25, 35J61, 35J20, 35B09, 35B32
1. Introduction and statements of main results
Let be a bounded domain of () with smooth boundary . This article is concerned with existence, non-existence, and multiplicity of non-negative solutions for the problem
is the usual Laplacian in ,
is the unit outer normal to the boundary .
By a solution of we mean a classical solution of . A solution of is said to be nontrivial and non-negative if it satisfies on and , whereas it is said to be positive if it satisfies on .
If and are positive on some non-empty open subset of then belongs to the class of concave-convex type nonlinearities. Since the work of Ambrosetti, Brezis and Cerami , this class of problems has been widely investigated, mostly for Dirichlet boundary conditions. In  the authors proved the existence of such that the problem
has a minimal positive solution for , at least one positive weak solution for , and no positive solution for [3, Theorem 2.1]. Moreover, if when then (1.1) has a second positive solution for [3, Theorem 2.3]. It was also proved that is the only positive solution of (1.1) which converges to in as [3, Theorem 2.2]. Most of the previous results were extended by De Figueiredo, Gossez, and Ubilla  to a larger class of concave-convex type problems, whose prototype is the analogue of for Dirichlet boundary conditions, i.e.
Several differences between and (1.2) may be observed. The most evident one arises in the definite case , with . It is known from [13, 14] that in this case (1.2) has a nontrivial non-negative solution for some . This result no longer holds for . As a matter of fact, if is a non-negative solution of then a simple integration provides
so that if .
The first purpose of this work is to obtain conditions on and which guarantee the existence of a nontrivial non-negative solution of for some . In particular, we shall obtain two nontrivial non-negative solutions for sufficiently small. At this point further differences between and (1.2) may be pointed out. Unlike [3, Theorem 2.2], we shall see that in some cases we have in as (see Theorem 1.1). Furthermore, in contrast with [3, 14], the second solution may be obtained without the condition when (see Remark 1.2).
To the best of our knowledge, very few works have been devoted to concave-convex problems under Neumann boundary conditions. Tarfulea  considered in the case , proving that is a necessary and sufficient condition for the existence of a positive solution. Making use of the sub-supersolutions method, the author proved the existence of such that problem has at least one positive solution for which converges to in as , and no positive solution for .
Garcia-Azorero, Peral, and Rossi  dealt with the problem
By means of a variational approach, they proved that if and when then there exists such that (1.3) has at least two positive solutions for , at least one positive solution for , and no positive solution for .
In , Alama investigated the problem
where and . Note that when this problem can be reduced to by a suitable rescaling. A special difficulty in this problem is the possible existence of dead core solutions when changes sign. Using variational, bifurcation, and sub-supersolutions techniques, the author proved existence, non-existence and multiplicity results for non-negative solutions in accordance with and . Moreover, these solutions are shown to be positive in the set where . However, the author did not discuss the structure of the non-negative solutions set when .
The second and main purpose of this article is to investigate the existence of a subcontinuum of non-negative solutions of . Some works have been devoted to this issue in the context of concave-convex nonlinearities. In , Korman proved that if is a ball in then there exists such that the problem
has exactly two positive solutions for , one positive solution for and no positive solution for . In addition, he proved that for the positive solutions lie on a single smooth solution curve and described the behavior of this curve with respect to . In  he extended these results to a problem with a non-autonomous concave-convex nonlinearity. Delgado and Suárez  considered the problem
where is a second order uniformly elliptic operator not necessarily self-adjoint and changes sign. They proved the existence of a unbounded subcontinuum of non-negative solutions emanating supercritically from . To the best of our knowledge, no results on the existence of a subcontinuum of non-negative solutions for are known when changes sign.
Based on the asymptotic analysis of as , we shall prove in some cases the existence of a loop type subcontinuum (see Theorem 1.6) in the non-negative solutions set of . This kind of continuum has been investigated by López-Gómez and Molina-Meyer in  and Brown in  for problems involving nonlinearities that are at , which is not the case for . For that same reason, the standard global bifurcation theory proposed by Rabinowitz  (see also López-Gómez ) does not apply to in a straightforward way. We shall overcome this difficulty using a regularization procedure that will be described later. Several works have made a direct use of the global bifurcation theory. We refer to Hess and Kato  for a problem with a non self-adjoint operator, to Blat and Brown  for a class of nonlinear elliptic systems, to López-Gómez and Molina-Meyer  for a study of isolas or compact solution components, to Cantrell and Cosner  for diffusive logistic equations from Mathematical Biology, and to Umezu  and Cano-Casanova  for nonlinear boundary conditions.
Note that if then, by the strong maximum principle and the boundary point lemma, nontrivial non-negative solutions of are positive solutions. On the other hand, it is known that if then dead core solutions may arise , which makes delicate the study of the non-negative solutions set of , as shown in . For instance, when changes sign the existence of a minimal non-negative solution for small is still unknown. Furthermore, when and changes sign, it is not known whether the condition provides non-existence of nontrivial non-negative solutions of for .
In our existence results we shall also be concerned with stability properties of positive solutions of . Let us recall that a positive solution of is said to be asymptotically stable (respect. unstable) if (respect. ), where is the first eigenvalue of the linearized problem at , namely,
In addition, is said to be weakly stable if .
Throughout this article, we consider the following sets:
Our main existence results for shall be obtained under the condition
If either or then we set
We are now in position to state out main results.
First we follow a variational approach to show that has two nontrivial non-negative solutions for small if (1.8) holds and . This approach also provides us with the asymptotic profiles of these solutions as :
Assume (1.8) and if . Then there exists such that:
If then has a nontrivial non-negative solution for . Moreover there holds in as . More precisely:
If, in addition, and then, up to a subsequence, in as , where is a nontrivial non-negative ground state solution of
If, in addition, then in as . In particular, if then is an asymptotically stable positive solution of for sufficiently small.
If then has a nontrivial non-negative solution for . Moreover there holds:
If, in addition, then and in as . In particular, is a unstable positive solution of for sufficiently small.
If, in addition, then in as .
If, in addition, and then, up to a subsequence, in as , where is a positive ground state solution of
In particular, is a unstable positive solution of for sufficiently small.
Except for (1)(a), 1(b) with , 2(b) and (2)(c), Theorem 1.1 remains true without the condition if . In the case and a solution having similar features as may be obtained by the sub-supersolutions method. Note that in contrast with the case of Dirichlet boundary conditions, obtaining a strict supersolution for is not an easy task. We shall use the asymptotic profile of provided by Theorem 1.1 (1)(a) to obtain such a supersolution, cf. Proposition 3.1. In the case we shall use the Lyapunov-Schmidt reduction method to obtain a positive solution such that in as . The same procedure can be applied in the case , cf. Remark 5.6.
When is a non-empty subdomain of and , we deduce from Theorem 1.1 (1)(a) that for any subset satisfying there exist such that for . This result comes from the fact that in .
Assume if . Let and be provided by Theorem 1.1.
If and then there exists such that on for .
If and then there exists such that on for .
As for non-existence of nontrivial non-negative solutions of , we have the following result:
Let . Then the following two assertions hold:
Assume and . Then has no nontrivial non-negative solution.
Assume that changes sign, is a subdomain of , and . If and then has no non-negative solution taking positive values somewhere in .
Assume . Then there exists such that has no nontrivial non-negative solution for .
Theorem 1.4 holds true for with replaced by . Indeed, it suffices to look at the equation in as .
We consider then structure of the non-negative solutions set of . Under the condition
Theorem 1.1 asserts that in as , and if then, up to a subsequence, in , where is a positive solution of (1.11). In addition, this result does not depend on the sign of . As a consequence we may also infer the existence of two nontrivial non-negative solutions and for sufficiently small. These solutions satisfy in as , and if then, up to a subsequence, in , where is a positive solution of (1.11). One may then ask if these solutions lie on a loop type subcontinuum of non-negative solutions of .
We shall investigate this question by considering a regularized version of , namely,
where and . We may then look at as the limit problem of when . This procedure has been already used in , where a regularized version of a nonlinear boundary condition is studied. Note that the mapping is analytic at and any nontrivial non-negative solution of is positive on . The unilateral global bifurcation theorem by Rabinowitz [24, Theorem 1.27] (see also López-Gómez [20, Theorem 6.4.3]) may then be applied to . To this end we consider its linearized problem at :
Under the condition
this problem has exactly two principal eigenvalues and , which are both simple.
We use the unilateral global bifurcation theory to obtain two subcontinua , of positive solutions of bifurcating from and , respectively. Moreover, we analyse the local nature of these subcontinua near the bifurcation points (Theorem 5.1).
We turn then to the study of the global nature of and their limiting nature as . First we show that positive solutions of are a priori bounded in if the following conditions are
assumed, where we assume following Amann and López-Gómez :
There exist balls , such that , and
are subdomains of with smooth boundary and satisfy , .
Under there exist a function which is continuous, positive, and bounded away from zero in a tubular neighborhood of in and such that
where denotes the distance function to a set . Moreover, we assume that
Based on these a priori bounds and the global properties of and , we infer that these subcontinua are both bounded, and consequently must coincide, i.e. (Theorem 6.7). Thus has a bounded subcontinuum of positive solutions going from to , see Figure 2. We consider then the limiting profiles of and as by means of Whyburn’s topological method . Here a priori bounds from below for positive solutions of with (Lemma 6.8) and the fact that bifurcation from zero does not occur for at any (Proposition 6.10) play an important role. The latter fact is verified under the condition
and are both subdomains of .
Combining the previous results, we establish:
Assume (1.12). If , , and are satisfied then has a loop type subcontinuum (non-empty, closed and connected component) of nontrivial non-negative solutions bifurcating at , which joins to itself. Moreover:
is non-trivial, i.e. .
The only trivial solution contained in is , i.e. does not contain any point with .
There exists such that does not contain any positive solution of (1.11) satisfying .
An example of satisfying conditions , and can be constructed as in Figure 1.
Finally, let us mention that our regularization procedure described above can also be used to obtain subcontinua (non-necessarily of loop type) for a larger class of concave-convex type problems. We shall treat this issue in a forthcoming article.
The outline of this article is the following: in Section 2 we follow a variational approach based on the Nehari manifold method to prove Theorem 1.1. In Section 3 we use the asymptotic profile of to obtain a nontrivial non-negative solution of for small via the sub-supersolutions method. We also show that bifurcation from zero does not occur for at any . Section 4 is devoted to the proof of Theorem 1.4. In Section 5 we carry out a bifurcation analysis for the regularized problem . Finally, in Section 6 we prove Theorem 1.6.
Throughout this article we use the following notations and conventions:
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 .
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 .
2. The variational approach
Throughout this section we assume that . We associate to the functional defined on by
Let us recall that is equipped with the usual norm . Critical points of are weak solutions of , which are also classical solutions by standard regularity. In the sequel we shall consider the following useful subsets of :
The next result will be used repeatedly in this section:
If is a sequence such that in and then is a constant and in .
Assume (respect. ). If and (respect. ) then is not a constant.
Since in and is weakly lower semicontinuous, we have
Hence , which implies that is a constant. Moreover . Since in we deduce that in .
If is a non-zero constant then , which is a contradiction.
The Nehari manifold associated to is given by
We shall use the splitting
Note that any nontrivial solution of belongs to . Furthermore, it follows from the implicit function theorem that is a manifold and every critical point of the restriction of to this manifold is a critical point of (see for instance [7, Theorem 2.3]), and therefore a solution of .
Note that any positive solution of belonging to is unstable. Indeed, if then
It follows that
To analyse the structure of , we consider the fibering maps corresponding to , which are set, for , as follows:
It is easy to see that
and more generally,
Having this characterisation in mind, we look for conditions under which has a critical point. Set
Let . Then has a global maximum at some , and moreover, is unique. If , then has a global maximum which is positive and a local minimum which is negative. Moreover, these are the only critical points of .
We shall require a condition on that provides . Note that
if and only if
if and only if
Note that satisfies for , i.e. is homogeneous of order .
We introduce now
Note that if then . We deduce then the following result, which provides sufficient conditions for the existence of critical points of :
Assume (1.8). Then and, for , there holds:
If either or then, has a positive global maximum at some , i.e. and for . Moreover, is the unique critical point of .
If either or then has a negative global minimum at some , i.e. and for . Moreover, is the unique critical point of .
If then has a negative local minimum at and a positive global maximum at . Furthermore and are the only critical points of .
First, we show that . Assume , so that we can choose satisfying
If is bounded in then we may assume that for some and in and . It follows from Lemma 2.1(1) that is a constant and in . From we deduce that . In addition, there holds
so that . From Lemma 2.1 we get a contradiction.
Let us assume now that . Set , so that . We may assume that and in . Since and , we have in ,