BIFURCATION OF POSITIVE SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS ROBIN AND NEUMANN PROBLEMS WITH COMPETING NONLINEARITIES

In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter λ > 0. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of λ > 0.

Hence a : R N → R N is a continuous and strictly monotone map, which satisfies certain other regularity and growth conditions, listed in hypotheses H(a) below. These conditions are general enough, to incorporate in our setting various differential operators of interest, such as the p-Laplacian (1 < p < ∞). Also, ∂u ∂n a denotes the conormal derivative defined by ∂u ∂n a = (a(Du), n) R N with n(z) being the outward unit normal at z ∈ ∂Ω. The reaction f (z, x, λ) is a parametric function with λ > 0 being the parameter and (z, x) → f (z, x, λ) is Carathéodory (that 5004 NIKOLAOS S. PAPAGEORGIOU AND VICENŢ IU D. RȂDULESCU is, for all x ∈ R the mapping z −→ f (z, x, λ) is measurable and for a.a. z ∈ Ω the map x −→ f (z, x, λ) is continuous). We assume that f (z, ·, λ) exhibits competing nonlinearities, namely near the origin, it has a "concave" term ( that is, a strictly (p − 1)-sublinear term), while near +∞, the reaction is "convex" term (that is, x −→ f (z, x, λ) is (p − 1)-superlinear). A special case of our reaction, is the following function: f (z, x, λ) = f (x, λ) = λx q−1 + x r−1 for all x ≥ 0, with 1 < q < p < r < p * = This reaction is encountered in the literature in the context of equations driven by the Laplacian (that is, p = 2) or by the p-Laplacian (1 < p < ∞).
Our aim is to investigate the existence, nonexistence and multiplicity of positive solutions as the parameter λ > 0 varies. So, we prove two bifurcation type results, describing the set of positive solutions of (P λ ) as the parameter λ > 0 changes, when the reaction exhibits the competing effects of concave (that is, sublinear) and convex (that is, superlinear) nonlinearities. In the first theorem the bifurcation occurs near zero. More precisely, under general hypotheses we show that there exists λ * > 0 such that the following properties hold: (a) for all λ ∈ (0, λ * ), problem (P λ ) has at least two positive solutions; (b) for λ = λ * problem (P λ * ) has at least one positive solution; (c) for all λ > λ * problem (P λ ) has no positive solution.
In the second case, we assume that β ≡ 0 (Neumann boundary condition) and we consider the problem      −div a(Du(z)) = f 0 (z, u(z)) − λu(z) p−1 in Ω, ∂u ∂n (z) = 0 on ∂Ω, We obtain a different geometry and we establish that the bifurcation occurs for large values of the parameter λ > 0. More precisely, under natural assumptions on f 0 we show that there exists λ * > 0 such that (a) for every λ > λ * problem (S λ ) has at least two positive solutions; (b) for λ = λ * problem (S λ * ) has at least one positive solution; (c) for every λ ∈ (0, λ * ) problem (S λ ) has no positive solution.
The first work concerning positive solutions for problems with concave and convex nonlinearities, was that of Ambrosetti, Brezis and Cerami [2]. They studied semilinear equations driven by the Dirichlet Laplacian and with a reaction of the form (1). Their work was extended to equations driven by the Dirichlet p-Laplacian by Garcia Azorero, Manfredi and Peral Alonso [10] and by Guo and Zhang [14]. We also refer to the contributions of de Figueiredo, Gossez and Ubilla [7], [8] to concave-convex type problems and general nonlinearities for the Laplacian, resp. p-Laplacian case. Extensions to equations involving more general reactions, were obtained by Gasinski and Papageorgiou [13], Hu and Papageorgiou [15] and Rȃdulescu and Repovš [22]. Other problems with competition phenomena, can be found in the works of Cîrstea, Ghergu and Rȃdulescu [4] (problems with singular terms) and of Kristaly and Moroşanu [16] (problems with oscillating reaction). Finally we mention the recent work of Papageorgiou and Rȃdulescu [20], who studied a Robin problem driven by the p-Laplacian and with a logistic reaction and proved multiplicity theorems for all large values of the parameter λ > 0, producing also nodal solutions.
We stress that the differential operator in (P λ ) is not homogeneous and this is a source of difficulties in the analysis of the problem, since many of the methods and techniques developed in the aforementioned papers do not work here. It appears that our results in the present paper are the first bifurcation-type theorems for nonhomogeneous elliptic equations.
2. Mathematical background. Let X be a Banach space and X * its topological dual. By ·, · we denote the duality brackets for the pair (X * , X). Given ϕ ∈ C 1 (X), we say that ϕ satisfies the Cerami condition (the C-condition), if the following is true: admits a strongly convergent subsequence". This is a compactness-type condition on the function ϕ which compensates for the fact that the space X need not be locally compact (being in general infinite dimensional). It is more general than the more common Palais-Smale condition. Nevertheless, the C-condition suffices to prove a deformation theorem and from it derive the minimax theory of the critical values of ϕ. One of the main results in that theory, is the so-called mountain pass theorem of Ambrosetti and Rabinowitz [3]. Here we state it in a slightly more general form.
Theorem 2.1. Let X be a Banach space, ϕ ∈ C 1 (X) satisfies the C-condition, u 0 , Then c ≥ m ρ and c is a critical value of ϕ.
Let η ∈ C 1 (0, ∞) and assume that The hypotheses on the map a(·) are the following: H(a) : a(y) = a 0 (|y|)y for all y ∈ R N , with a 0 (t) > 0 for all t > 0 and (ii) |∇a(y)| ≤ c 3 η(|y|) |y| for some c 3 > 0, all y ∈ R N \{0}; Remark 1. These conditions on a(·) are motivated by the regularity results of Lieberman [17] and the nonlinear maximum principle of Pucci and Serrin [21]. According to the above conditions, the potential function G 0 (·) is strictly convex and strictly increasing. We set G(y) = G 0 (|y|) for all y ∈ R N . Then the function y −→ G(y) is convex and differentiable on R N \{0}. We have So, G(·) is the primitive of the map a(·). Because G(0) = 0 and y −→ G(y) is convex, from the properties of convex functions, we have The next lemma summarizes the main properties of the map a(·). They follow easily from hypotheses H(a) above.
(a) y −→ a(y) is continuous and strictly monotone, hence maximal monotone too; Lemma 2.2 together with (1) and (2), lead to the following growth estimates for the primitive G(·).
In the analysis of problem (P λ ) in addition to the Sobolev space W 1,p (Ω), we will also use the Banach space C 1 (Ω). This is an ordered Banach space, with positive cone C + = {u ∈ C 1 (Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior given by int C + = {u ∈ C + : u(z) > 0 for all z ∈ Ω}.
To distinguish, we use | · | to denote the norm of R N . If on ∂Ω we use the (N − 1)-dimensional Hausdorff measure σ(·) (the surface measure on ∂Ω), then we can define the Lebesgue spaces L q (∂Ω), 1 ≤ q ≤ ∞. We know that there exists a unique continuous, linear map γ 0 : W 1,p (Ω) → L p (∂Ω), known as the trace map, such that γ 0 (u) = u| ∂Ω for all u ∈ C 1 (Ω). In fact γ 0 is compact. We have In the sequel, for the sake of notational simplicity, we drop the use of the trace map γ 0 , with the understanding that all restrictions of elements of W 1,p (Ω) on ∂Ω, are defined in the sense of traces.
Suppose f 0 : Ω × R → R is a Carathéodory function with subcritical growth in the x ∈ R variable, that is |f 0 (z, x)| ≤ a 0 (z)(1 + |x| r−1 ) for a.a. z ∈ Ω, all x ∈ R, with a 0 ∈ L ∞ (Ω) + , 1 < r < p * . We set F 0 (z, x) = x 0 f 0 (z, s)ds and consider the The next proposition, was proved by Papageorgiou and Rȃdulescu [20] for G(y) = 1 p |y| p for all y ∈ R N . The proof remains valid in the present more general setting, using Lemma 2.2, Corollary 1 and the regularity result of Lieberman [17] [p. 320].

NIKOLAOS S. PAPAGEORGIOU AND VICENŢ IU D. RȂDULESCU
Let A : W 1,p (Ω) → W 1,p (Ω) * be the nonlinear map defined by The following, is a particular case of a more general result due to Gasinski and Papageorgiou [12].
3. Bifurcation near zero for the Robin problem. In this section, we deal with competition phenomena that give rise to bifurcation of the problem solutions, when the parameter λ > 0 is near zero. This situation includes the classical equations with concave and convex nonlinearities.
So, we see that the AR-condition implies hypothesis H 1 (iv). This weaker "superlinearity" condition, incorporates in our setting (p−1)-superlinear nonlinearities with "slower" growth near +∞, which fail to satisfy the AR-condition (see the examples below). Finally note that hypothesis H 1 (v) implies the presence of a concave nonlinearity near zero.
Example 2. The following functions satisfy hypotheses H 1 . For the sake of simplicity, we drop the z-dependence: Note that f 2 (·, λ) does not satisfy the AR-condition. We introduce the following Carathéodory function (z, s, λ)ds and consider the C 1 -functionalφ λ : Proposition 3. If hypotheses H(a), H(β) and H 1 hold and λ > 0, then the functionalφ λ satisfies the C-condition.
Then the previous argument works and leads again to (19). From (9) and (19) it follows that {u n } n≥1 ⊆ W 1,p (Ω) is bounded. So, we may assume that u n w → u in W 1,p (Ω) and u n → u in L r (Ω) and in L p (∂Ω).
We can show that S is nonempty, as well as a useful structural property of the solution set S(λ). Proposition 6. If hypotheses H(a), H(β) and H 1 hold, then S = ∅ and for every λ ∈ S ∅ = S(λ) ⊆ int C + .
Next, we consider the following truncation-perturbation of the reaction in problem (P η ): This is a Carathéodory function. Let Γ η (z, x) = x 0 γ η (z, s)ds and consider the for all u ∈ W 1,p (Ω).
Note that Moreover, Proposition 5 implies that if u ∈ int C + , then Claim 1. We may assume that u 0 is a local minimizer of σ η .
This proves the Claim. Reasoning as above, we can show that Then from (40) we see that the elements of K ση are positive solutions of problem (P η ). Therefore, we may assume that K ση is finite of otherwise we already have an infinity of positive solutions for problem (P η ).
Next we examine what happens in the critical case λ = λ * . To this end, note that hypotheses H 2 (ii), (v) imply that we can find c 18 = c 18 (λ) > 0 such that This unilateral growth estimate on the reaction f (z, ·, λ) leads to the following auxiliary Robin problem: For this problem we have the following existence and uniqueness result. Proof. First we show the existence of a positive solution for problem (49). To this end let ξ + : W 1,p (Ω) → R be the C 1 -functional defined by for all u ∈ W 1,p (Ω).
Then e(·) is Gâteaux differentiable atū τ andȳ τ in the direction h. Moreover, via the chain rule and the nonlinear Green's identity, we obtain (recall that C 1 (Ω) is dense in W 1,p (Ω)). The convexity of e implies the monotonicity of e . Then This proves the uniqueness of the positive solutionū ∈ int C + of problem (49).
Proof. Let u ∈ S(λ) and consider the following Carathéodory function: Let W (z, x) = x 0 w(z, s)ds and consider the C 1 -functionalγ : for all u ∈ W 1,p (Ω).
Now we can show that the critical value λ * is admissible, that is λ * ∈ S.
From (57), as in the proof of Proposition 3, using hypothesis H 2 (iv), we show that {u n } n≥1 ⊆ W 1,p (Ω) is bounded. So, we may assume that u n w → u * in W 1,p (Ω) and u n → u * in L r (Ω) and in L p (∂Ω) as n → ∞.
Remark 5. Note that in the above proof we have proved that, if λ ∈ S 0 , u λ ∈ S 0 (λ) ⊆ int C + and µ > λ, then µ ∈ S 0 and we can find u µ ∈ S 0 (µ) ⊆ int C + such that u µ ≤ u λ . In fact in the next proposition, we improve this conclusion.
Again the difficulty is in Proposition 15. The proof of that proposition fails since we can not control the boundary term ∂Ω β(z)u p−1 n dσ.
Thus we may assume that u n w →ŵ λ in W 1,p (Ω) and u n →ŵ Λ in L r (Ω).
Proof. Note that A(ŵ λ ), h + µ We consider the problem (S µ ) and truncate the reaction f 0 (z, ·) at {0,ŵ λ (z)}. Then reasoning as in the proof of Proposition 13, via the direct method and using