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###### Abstract

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 .

Indefinite potential, Robin boundary condition, strong maximum principle, truncation, competing nonlinear, positive solutions, regularity theory, minimal positive solution.

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)

## 1 Introduction

Let () be a bounded domain with a -boundary . In this paper we study the following parametric Robin problem

 ⎧⎪⎨⎪⎩−Δu(z)+ξ(z)u(z)=λg(z,u(z))+f(z,u(z)) in Ω∂u∂n+β(z)u=0 on ∂Ω.⎫⎪⎬⎪⎭ (Pλ)

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 [2], who dealt with a Dirichlet problem with zero potential (that is, ) and the reaction had the form

 λxq−1+xr−1 for all x≥0 with 1

They proved a bifurcation-type result for small values of the parameter . The work of Ambrosetti, Brezis and Cerami [2] was extended to more general classes of Dirichlet problems with zero potential by Bartsch and Willem [4], Li, Wu and Zhou [9], and Rădulescu and Repovš [19].

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 [13] 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 .

## 2 Preliminaries

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

 (1+||un||)φ′(un)→0 in X∗ as n→∞,

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 [6]). The following notion is central to this theory.

###### Definition 2.1.

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]).

###### Theorem 2.2.

Assume that is a Banach space, are nonempty, closed subsets such that is linking with in , satisfies the -condition

 supE0φ

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”.

###### Theorem 2.3.

Assume that is a Banach space, satisfies the -condition, ,

 max{φ(u0),φ(u1)}

and with . Then and is a critical value of .

###### Remark 1.

Theorem 2.3 can be deduced from Theorem 2.2 if we have , ,

In the analysis of problem (), we will use the following spaces: the Sobolev space , the Banach space and the boundary Lebesgue spaces , .

By we denote the norm of the Sobolev space . So

 ||u||=[||u||22+||Du||22]12 for all u∈H1(Ω).

The space is an ordered Banach space with positive cone

 C+={u∈C1(¯¯¯¯Ω): u(z)≥0 for all z∈¯¯¯¯Ω}.

We will use the open set defined by

 D+={u∈C+:u(z)>0 for all z∈¯¯¯¯Ω}.

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

 γ0(u)=u|∂Ω for all u∈H1(Ω)∩C(¯¯¯¯Ω).

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

 |f0(z,x)|≤a0(z)(1+|x|r−1) for almost all z∈Ω and all x∈R,

with

We set . Also, let and with on . We consider the -functional defined by

 φ0(u)=12ϑ(u)−∫ΩF0(z,u)dz,

where

 ϑ(u)=||Du||22+∫Ωξ(z)u2dz+∫∂Ωβ(z)u2dσ for all u∈H1(Ω).

The next result follows from Papageorgiou and Rădulescu [12, Proposition 3] using the regularity theory of Wang [20].

###### Proposition 1.

Let be a local -minimizer of , that is, there exists such that

 φ0(u0)≤φ0(u0+h) for all h∈C1(¯¯¯¯Ω) with ||h||C1(¯¯¯Ω)≤ρ0.

Then with and is also a local -minimizer of , that is, there exists such that

 φ0(u0)≤φ0(u0+h) for all h∈H1(Ω) with ||h||≤ρ1.

We will need some facts concerning the spectrum of with Robin boundary condition. Details can be found in Papageorgiou and Rădulescu [12, 16].

So, we consider the following linear eigenvalue problem

 −Δu(z)+ξ(z)u(z)=^λu(z) in Ω, ∂u∂n+β(z)u=0 on ∂Ω. (1)

We know that there exists such that

 ϑ(u)+μ||u||22≥c0||u||2 for all u∈H1(Ω) and for some c0>0. (2)

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 [20] implies that . By (for every ) we denote the eigenspace corresponding to the eigenvalue . We have the following orthogonal direct sum decomposition

 H1(Ω)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⊕k≥1E(^λk).

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:

 ∙ ^λ1 is simple (that is, dimE(^λ1)=1); ∙ ^λ1=inf[ϑ(u)||u||22:u∈H1(Ω),u≠0]; ∙ for m≥2 we have
 ^λm= sup[ϑ(u)||u||22:u∈⊕mk=1E(^λk),u≠0] = inf[ϑ(u)||u||22:u∈¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⊕k≥mE(^λk),u≠0] (4)

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.

###### Proposition 2.
• If then .

• If then .

Finally, let us fix some basic notations and terminology. So, by
we denote the linear operator defined by

 ⟨A(u),h⟩=∫Ω(Du,Dh)RNdz for all u,h∈H1(Ω).

A Banach space is said to have the “Kadec-Klee property” if the following holds

 ‘‘un\lx@stackrelw→u in X and%  ||un||→||u||⇒un→u in X".

Locally uniformly convex Banach spaces, in particular Hilbert spaces, have the Kadec-Klee property.

Let . We set and for we define

 u±(⋅)=u(⋅)±.

We know that

 u±∈H1(Ω), |u|=u++u−, u=u+−u−.

By we denote the Lebesgue measure on . Also, if then

 Kφ={u∈X:φ′(u)=0} (the critical % set of φ).

If , then and . Finally, we set

 n0=max{k∈N:^λk≤0}.

If for all (this is the case if and or ), then we set .

## 3 Positive solutions

The hypotheses on the data of problem () are the following:

.

.

• for every , there exists such that

 g(z,x)≤aρ(z) for almost all z∈Ω and all x∈[0,ρ];
• uniformly for almost all ;

• there exist constants and such that

 c3xq−1≤g(z,x) for almost all z∈Ω and all x≥0,
 limsupx→0+g(z,x)xq−1≤c4 % uniformly for almost all z∈Ω;
• if , then  for almost all and all ;

• for every , there exists such that for almost all the function

 x↦g(z,x)+^ξρx

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

 f(z,x)≥0 for almost all z∈Ω and all x∈[0,δ0];
• for every , there exists such that for almost all the function

 x→f(z,x)+~ξρx

is nondecreasing on

We set and define

 γλ(z,x)=λg(z,x)+f(z,x)−2[λG(z,x)+F(z,x)] % for all (z,x)∈Ω×R+.

For every , there exists such that

 γλ(z,x)≤γλ(z,y)+eλ(z) for % almost all z∈Ω and all 0≤x≤y.
###### Remark 2.

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

 g(z,x)=f(z,x)=0 for almost all z∈Ω all x≤0

(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 [3]). Instead, we use hypothesis , which is a slightly more general version of a condition used by Li and Yang [10]. 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 ,

 x↦λg(z,x)+f(z,x)x

is nondecreasing on (see [10]).

Examples. The following pair satisfies hypotheses and :

 g(z,x)=a(z)xq−1, f(z,x)=b(z)xr−1 for all x≥0

with for almost all and . If , this is the reaction pair used by Ambrosetti, Brezis and Cerami [2] in the context of Dirichlet problems with zero potential (that is, ). The above reaction pair was used by Rădulescu and Repovš [19], 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):

 g(x)={2xq−1−xτ−1if 0≤x≤1xη−1if 1

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

 kλ(z,x)=λg(z,x)+f(z,x)+μx+. (5)

We set and consider the -functional defined by

 ^φλ(u)=12ϑ(u)+μ2||u||22−∫ΩKλ(z,u)dz for all u∈H1(Ω).
###### Proposition 3.

If hypotheses and hold, then for every the functional satisfies the C-condition.

###### Proof.

Let be a sequence such that

 |^φλ(un)|≤M1 for some M1>0 and all n∈N, (6) (1+||un||)^φ′λ(un)→0 in H1(Ω)∗ as n→∞. (7)

By (7) we have

 ∣∣∣⟨A(un),h⟩+∫Ω(ξ(z)+μ)unhdz+∫∂Ωβ(z)unhdσ−∫Ωkλ(z,un)hdz∣∣∣≤ϵn||h||1+||un||, (8)

for all with .

In (8) we choose . Then

 ϑ(u−n)+μ||u−n||22≤ϵn for all n∈N (see (???)), (9) ⇒ c0||u−n||2≤ϵn for all n∈N (see (???)), ⇒ u−n→0 in H1(Ω).

It follows from (6) and (9) that

 ϑ(u+n)−∫Ω2[λG(z,u+n)+F(z,u+n)]dz≤M2 for some M2>0 and all n∈N. (10)

If in (8) we choose , then

 −ϑ(u+n)+∫Ω[λg(z,u+n)+f(z,u+n)]u+ndz≤ϵn for all n∈N. (11)

Adding (10) and (11), we obtain

 ∫Ωγλ(z,u+n)dz≤M3 for some M3>0 and all n∈N. (12)

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

 ||yn||=1, yn≥0 for all n∈N

and so we may assume that

 yn\lx@stackrelw→y in H1(Ω) and yn→y in L2s′(Ω) % and in L2(∂Ω),y≥0. (13)

Suppose that and let . Then and

 u+n(z)→+∞ for almost all z∈Ω∗.

We have

 G(z,u+n)||u+n||2=G(z,u+n)(u+n)2y2n→0 for a.a. z∈Ω∗ (% see hypothesis H(g)(ii)), (14) F(z,u+n)||u+n||2=F(z,u+n)(u+n)2y2n→+∞ for a.a. z∈Ω∗ (see hypothesis H(f)(ii)). (15)

It follows from (14), (15) and Fatou’s lemma that

 limn→∞[λ∫ΩG(z,u+n)||u+n||2dz+∫ΩF(z,u+n)||u+n||2dz]=+∞. (16)

On the other hand, (6) and (9) imply that

 λ∫ΩG(z,u+n)||u+n||2dz+∫ΩF(z,u+n)||u+n||2dz≤M1||u+n||2+12ϑ(yn)+μ2||yn||22≤M4 (17)

for some , all .

Comparing (16) and (17) we obtain a contradiction.

Next, suppose that . For we set . Then in and so we have

 ∫ΩG(z,^yn)dz→0 and ∫ΩF(z,^yn)dz→0. (18)

Since , we can find such that

 (2η)12||u+n||∈(0,1] for all n≥n1. (19)

We choose such that

 ^φλ(tnu+n)=max[^φλ(tu+n):0≤t≤1]. (20)

From (19), (20) we have

 ^φλ(tnu+n)≥ ^φλ(^yn) (see (???)) = 12ϑ(^yn)−λ∫ΩG(z,^yn)dz−∫ΩF(z,^yn)dz (see (???)) ≥ η−λ∫ΩG(z,^yn)dz−∫ΩF(z,^yn)dz ≥ 12η for all n≥n2≥n1 (% see (???)). (21)

Since is arbitrary, we infer from (3) that

 ^φλ(tnu+n)→+∞ as n→∞. (22)

We know that

 ^φλ(0)=0 and ^φλ(u+n)≤M5 for some M5>0 and all n∈N (see (???) and (???)), ⇒ tn∈(0,1) for all n≥n3 (see (???)).

So, (20) implies that

 tnddt^φλ(tu+n)|t=tn=0, (23) ⇒ ⟨^φ′λ(tnu+n),tnu+n⟩=0 (by the chain rule), ⇒ ϑ(tnu+n)=∫Ω[λg(z,tnu+n)+f(z,tnu+n)](tnu+n)dz for all n≥n3.

We have . Then hypothesis