We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies. We also show the existence of a minimal positive solution and determine the monotonicity and continuity properties of the map .
Nonlinear Robin problems] Robin problems with indefinite linear part and competition phenomena N.S. Papageorgiou, V.D. Rădulescu and D.D. Repovš] \subjclassPrimary: 35J20, 35J60; Secondary: 35J92. ††thanks: Corresponding author: Vicenţiu D. Rădulescu
Nikolaos S. Papageorgiou
Department of Mathematics, National Technical University
Zografou Campus, Athens 15780, Greece
Vicenţiu D. Rădulescu
Department of Mathematics, Faculty of Sciences,
King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Department of Mathematics, University of Craiova,
Street A.I. Cuza No. 13, 200585 Craiova, Romania
Dušan D. Repovš
Faculty of Education and Faculty of Mathematics and Physics,
University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia
(Communicated by Xuefeng Wang)
Let () be a bounded domain with a -boundary . In this paper we study the following parametric Robin problem
In this problem, is a parameter, () is a potential function which is indefinite (that is, sign changing) and in the reaction, and are Carathéodory functions (that is, for all , are measurable and for almost all , are continuous). We assume that for almost all , is strictly sublinear near (concave nonlinearity), while for almost all , is strictly superlinear near (convex nonlinearity). Therefore the reaction in problem () exhibits the combined effects of competing nonlinearities (“concave-convex problem”). The study of such problems was initiated with the well-known work of Ambrosetti, Brezis and Cerami , who dealt with a Dirichlet problem with zero potential (that is, ) and the reaction had the form
They proved a bifurcation-type result for small values of the parameter . The work of Ambrosetti, Brezis and Cerami  was extended to more general classes of Dirichlet problems with zero potential by Bartsch and Willem , Li, Wu and Zhou , and Rădulescu and Repovš .
Our aim in this paper is to extend all the aforementioned results to the more general problem (). Note that when , we recover the Neumann problem with an indefinite potential. Robin and Neumann problems are in principle more difficult to deal with, due to the failure of the Poincaré inequality. Therefore in our problem, the differential operator (left-hand side of the equation) is not coercive (unless , ). Recently we have examined Robin and Neumann problems with indefinite linear part. We mention the works of Papageorgiou and Rădulescu [13, 14, 16]. In  the problem is parametric with competing nonlinearities. The concave term is , , (so it enters into the equation with a negative sign) while the perturbation is Carathéodory, asymptotically linear near and resonant with respect to the principal eigenvalue. We proved a multiplicity result for all small values of the parameter , producing five nontrivial smooth solutions, four of which have constant sign (two positive and two negative).
In this paper, using variational tools together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem, describing the existence and multiplicity of positive solutions as the parameter varies. We also establish the existence of a minimal positive solution and determine the monotonicity and continuity properties of the map .
Let be a Banach space and its topological dual. By we denote the duality brackets for the dual pair . Given , we say that satisfies the “Cerami condition” (the “C-condition” for short), if the following property is satisfied:
“Every sequence such that is bounded and
admits a strongly convergent subsequence”.
This is a compactness-type condition on the functional . It leads to a deformation theorem from which one can derive the minimax theory for the critical values of (see, for example, Gasinski and Papageorgiou ). The following notion is central to this theory.
Let be a Hausdorff topological space and nonempty, closed sets such that . We say that the pair is linking with in if:
For any such that , we have .
Using this topological notion, one can prove the following general minimax principle, known in the literature as the “linking theorem” (see, for example, Gasinski and Papageorgiou [6, p. 644]).
Assume that is a Banach space, are nonempty, closed subsets such that is linking with in , satisfies the -condition
and , where . Then and is a critical value of (that is, there exists such that ).
With a suitable choice of the linking sets, we can produce as corollaries of Theorem 2.2, the main minimax theorems of the critical point theory. For future use, we recall the so-called “mountain pass theorem”.
Assume that is a Banach space, satisfies the -condition, ,
and with . Then and is a critical value of .
By we denote the norm of the Sobolev space . So
The space is an ordered Banach space with positive cone
We will use the open set defined by
On we consider the -dimensional Hausdorff (surface) measure
Using this measure, we can define the Lebesgue spaces () in the usual way. Recall that the theory of Sobolev spaces says that there exists a unique continuous linear map , known as the “trace map”, such that
This map is not surjective and it is compact into if and into
In what follows, for the sake of notational simplicity, we drop the use of the map . All restrictions of Sobolev functions on are understood in the sense of traces.
Let be a Carathéodory function such that
We set . Also, let and with on . We consider the -functional defined by
Let be a local -minimizer of , that is, there exists such that
Then with and is also a local -minimizer of , that is, there exists such that
So, we consider the following linear eigenvalue problem
We know that there exists such that
Using (2) and the spectral theorem for compact self-adjoint operators, we generate the spectrum of (1), which consists of a strictly increasing sequence such that . Also, there is a corresponding sequence of eigenfunctions which form an orthonormal basis of and an orthogonal basis of . In fact, the regularity theory of Wang  implies that . By (for every ) we denote the eigenspace corresponding to the eigenvalue . We have the following orthogonal direct sum decomposition
Each eigenspace has the so-called “unique continuation property” (UCP for short) which says that if vanishes on a set of positive Lebesgue measure, then . The eigenvalues have the following properties:
In (2) the infimum is realized on .
In (2) both the supremum and the infimum are realized on .
From these properties, it is clear that the elements of have constant sign while for the elements of are nodal (that is, sign changing). Let denote the -normalized (that is, ) positive eigenfunction corresponding to . As we have already mentioned, . Using Harnack’s inequality (see, for example Motreanu, Motreanu and Papageorgiou [11, p. 212]), we have that for all . Moreover, if , then using the strong maximum principle, we have .
The following useful inequalities are also easy consequences of the above properties.
If then .
If then .
Finally, let us fix some basic notations and terminology. So, by
we denote the linear operator defined by
A Banach space is said to have the “Kadec-Klee property” if the following holds
Locally uniformly convex Banach spaces, in particular Hilbert spaces, have the Kadec-Klee property.
Let . We set and for we define
We know that
By we denote the Lebesgue measure on . Also, if then
If , then and . Finally, we set
If for all (this is the case if and or ), then we set .
3 Positive solutions
for every , there exists such that
uniformly for almost all ;
there exist constants and such that
if , then for almost all and all ;
for every , there exists such that for almost all the function
is nondecreasing on .
is a Carathéodory function such that
for almost all and all with ;
uniformly for almost all ;
uniformly for almost all and there exists such that
for every , there exists such that for almost all the function
is nondecreasing on
We set and define
For every , there exists such that
Since we are looking for positive solutions and all of the above hypotheses concern the positive semi-axis , we may assume without any loss of generality that
(note that hypotheses and imply that for almost all ). Hypothesis implies that for almost all is strictly sublinear near . This, together with hypothesis , implies that is globally the “concave” contribution to the reaction of problem (). On the other hand, hypothesis implies that for almost all is strictly superlinear near . Hence is globally the “convex” contribution to the reaction of (). Therefore on the right-hand side (reaction) of problem (), we have the competition of concave and convex nonlinearities (“concave-convex problem”). We stress that the superlinearity of is not expressed using the well-known Ambrosetti-Rabinowitz condition (see Ambrosetti and Rabinowitz ). Instead, we use hypothesis , which is a slightly more general version of a condition used by Li and Yang . Hypothesis is less restrictive than the Ambrosetti-Rabinowitz superlinearity condition and permits the consideration of superlinear terms with “slower” growth near , which fail to satisfy the AR-condition (see the examples below). Hypothesis is a quasimonotonicity condition on and it is satisfied if there exists such that for almost all ,
is nondecreasing on (see ).
Examples. The following pair satisfies hypotheses and :
with for almost all and . If , this is the reaction pair used by Ambrosetti, Brezis and Cerami  in the context of Dirichlet problems with zero potential (that is, ). The above reaction pair was used by Rădulescu and Repovš , again for Dirichlet problems with .
Another possibility of a reaction pair which satisfies hypotheses and are the following functions (for the sake of simplicity, we drop the -dependence):
In this pair, the superlinear term fails to satisfy the Ambrosetti-Rabinowitz condition.
Let be as in (2) and . Let be the Carathéodory function defined by
We set and consider the -functional defined by
If hypotheses and hold, then for every the functional satisfies the C-condition.
Let be a sequence such that
By (7) we have
for all with .
In (8) we choose . Then
If in (8) we choose , then
Claim. is bounded.
We argue by contradiction. So, suppose that the claim is not true. By passing to a subsequence if necessary, we may assume that . Let , . Then
and so we may assume that
Suppose that and let . Then and
Next, suppose that . For we set . Then in and so we have
Since , we can find such that
We choose such that
Since is arbitrary, we infer from (3) that
We know that
So, (20) implies that
We have . Then hypothesis