POSITIVE SOLUTIONS FOR RESONANT ( p, q ) -EQUATIONS WITH CONCAVE TERMS

. We consider a parametric ( p,q )-equation with competing nonlin-earities in the reaction. There is a parametric concave term and a resonant Caratheordory perturbation. The resonance is with respect to the principal eigenvalue and occurs from the right. So the energy functional of the problem is indeﬁnite. Using variational tools and truncation and comparison techniques we show that for all small values of the parameter the problem has at least two positive smooth solutions.

In problem (P λ ) we have the sum of a p-Laplacian and a q−Laplacian and this differential operator is nonhomogeneous. This is a source of difficulties in the analysis of the problem. Boundary value problems with a combination of differential operators of different nature (such as the (p, q)-equation above), arise in the study of various physical phenomena such as reaction-diffusion equations (see Cherfils-Ilyason [5]), particle physics (see Benci-Fortunato-Pisani [3]) and plasma physics (see Wilhelmsson [31]). The potential function ξ ∈ L ∞ (Ω) and ξ(z) ≥ 0 for a.a. z ∈ Ω. In the right hand side (forcing term) we have the competing effects of two nonlinearties, one is a parametric concave term while the other is a Caratheodory perturbation f . Namely, z → f (z, x) is measurable and x → f (z, x) is continuous.
Here we look for positive solutions for (p, q)-equations in which the reaction exhibits the competing effects of a concave term and a resonant term. The resonance is with respect to the principal eigenvalue λ 1 (p, ξ, β) > 0 of the differential operator u → −∆ p u+ξ(z)|u| p−2 u with the Robin boundary condition ∂u ∂npq +β(z)|u| p−2 u = 0 on ∂Ω. The resonance occurs from the right of λ 1 in the sense that This makes the energy functional ϕ λ of the problem (P λ ) indefinite (unbounded from above and below) and so we can not use the direct method of the calculus of variations directly on ϕ λ . We use instead variational methods based on the critical point theory together with suitable truncation and comparison techniques. We show that for all λ > 0 small, (P λ ) has two positive smooth solutions.
2. Preliminaries -hypotheses. Let X be a Banach space and X * its dual. ·, · denotes the duality brackets for (X * , X). Given ϕ ∈ C 1 (X, R), we say that ϕ satisfies the "Cerami condition" (the C-condition for short) if "every sequence {x n } ⊆ X such that {ϕ(x n )} is bounded and (1 + x n )ϕ (x n ) → 0 in X * , admits a strongly convergent subsequence". This is a compactness-type condition on ϕ that compensates for the fact that the ambient space X may not be locally compact. The C-condition is more general than the usual Palais-Smale condition (PS-condition for short). Nevertheless, the C-condition suffices to prove a deformation theorem and then from it we can derive the minimax theory of certain critical values of ϕ. One such minimax theorem which will be used shortly is the so-called mountain pass theorem. See, for example, Gasinski-Papageorgiou [13, p.648].
then c ≥ η r and c is a critical value of ϕ.
In the study of problem (P λ ) we will use the Sobolev space W 1,p (Ω), the space C 1 (Ω) and the boundary Lebesgue spaces L r (∂Ω), 1 ≤ r ≤ ∞. For u ∈ W 1,p (Ω), define u = u p p + Du p p 1/p .
The Banach space C 1 (Ω) is an ordered Banach space with positive cone This cone has nonempty interior given by Note that if D + = u ∈ C + : u(z) > 0 for all z ∈ Ω , then D + is open in C 1 (Ω) and D + ⊆ int C + . In fact D + is the interior of C + when C 1 (Ω) is equipped with the relative C(Ω)-norm topology.
On ∂Ω we consider the (N −1)-dimensional Hausdorff (surface) measure σ. Using σ(·) we can define in the usual way the boundary Lebesgue spaces L r (∂Ω). 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 ∈ W 1,p (Ω) ∩ C( Ω). So the trace map extends the notion of boundary values to all Sobolev functions. We know that im γ 0 = W 1 p ,p (∂Ω) and ker γ 0 = W 1,p 0 (Ω), where 1 p + 1 p = 1. Also, we know that γ 0 is compact into L r (∂Ω) for all r ∈ [1, (N −1)p N −p ) if p < N , and into L r (∂Ω) for all 1 ≤ r < ∞ if N ≤ p. In the sequel for the sake of notational simplicity we drop the use of the trace map. All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces.
For every r ∈ (1, ∞) let A r : W 1,r (Ω) → W 1,r (Ω) * be the nonlinear map defined by for all u, h ∈ W 1,r (Ω). The next proposition summarizes the main properties of this map (see Motreanu-Motreanu-Papageorgiou [22, p.40]. Proposition 2. The map A r is bounded, continuous, monotone (hence maximal monotone too), and of type (S) + , namely we have u n → u in W 1,p (Ω), provided that u n w → u in W 1,p (Ω) and lim sup n→∞ A(u n ), u n − u ≤ 0.
Remark. The above hypotheses include the case β ≡ 0, which corresponds to the Neumann problem.
From Mugnai-Papageorgiou [21,Lemma 4.11] we have Proposition 3. If H(ξ) holds and ξ = 0, then there exists c 0 > 0 such that for u ∈ W 1,p (Ω), Also, from Gasinski-Papageorgiou [13, Proposition 2.4] we have In the sequel we let µ p : Consider a Caratheodory function We set F 0 (z, x) = x 0 f 0 (z, s)ds and consider the C 1 -functional ϕ 0 : The next proposition is a particular case of a more general result of [24, Proposition 8]. The result is an outgrowth of the nonlinear regularity theory of Lieberman [18].
As we already mentioned in the Introduction, our approach involves comparison techniques. So, we will need the following strong comparison principle, due to Papageorgriou-Radulescu-Repovs [27,Proposition 7].

POSITIVE SOLUTIONS FOR RESONANT (p, q)-EQUATIONS WITH CONCAVE TERMS2643
We say that λ is an eigenvalue if problem (1) admits a nontrivial solution u ∈ W 1,p (Ω), known as an eigenfunction corresponding to the eigenvalue λ. The nonlinear regularity theory implies that every eigenfunction is in C 1 (Ω). Let σ denote the set of all the eigenvalues. Problem (1) was studied by Mugnai-Papageorgiou [21] (Neumann problem) and Fragnelli-Mugnai-Papageorgiou [10], Papageorgiou-Radulescu [24] (Robin problem). The next proposition states some of their results.
every eigenfunction corresponding to an eigenvalue other than λ 1 is nodal, i.e., sign changing.
The above properties imply that the elements of the principal eigenspace do not change sign. Let u 1 denote the positive L p -normalized eigenfunction corresponding to λ 1 > 0. From the strong nonlinear maximum principle (see [13, Proposition 6.2.8, p.738] and Pucci-Serrin [30, pp.111, 120], we have that u 1 ∈ D + . From (2) and Propositions 3 and 4 it is clear that under H(ξ), H(β) and H 0 we have λ 1 > 0. In (2) the infimum is realized in the corresponding one dimensional space.
We also have a weighted version of the eigenvalue problem (1). So let 0 ≤ m ∈ L ∞ (Ω) with m = 0, and consider Again we have a smallest eigenvalue λ 1 (m) > 0, which has the same properties as λ 1 . In this case the variational characterization of λ 1 takes the following form: As before the infimum is realized on the corresponding one dimensional eigenspace, whose elements have constant sign. Now let u 1 (m) denote the positive L pnormalized eigenfunction corresponding to λ 1 (m) > 0. We have u 1 (m) ∈ D + . These properties lead to the following monotonicity property of the map m → λ 1 (m).
and the two inequalities are strict on sets of positive measure in Ω, then λ 1 (m 2 ) < λ 1 (m 1 ).
Next we introduce the conditions on the perturbation term f .
H(f )(ii) implies that at +∞ we can have resonance with respect to the principal eigenvalue λ 1 > 0. As we shall see in the process of the proof, H(f )(iii) implies that the resonance occurs from the right of λ 1 in the sense that pF (z, x) − λ 1 x p → +∞ uniformly for a.a. z ∈ Ω as x → +∞.
This makes the energy functional unbounded below. In H(f )(iv) the inequality in W 1,p (Ω) * means that for all 0 ≤ h ∈ W 1,p (Ω). This assumption is satisfied if ξ = 0 and there are t > 0 and η 0 > 0 such that Note that no asymptotic condition is assumed as x → 0 + .

POSITIVE SOLUTIONS FOR RESONANT (p, q)-EQUATIONS WITH CONCAVE TERMS2645
Example. The following function satisfies H(f ). For the sake of simplicity we dropped the z-dependence.
3. An auxiliary Robin Problem. In this section we examine the following auxiliary nonlinear parametric Robin problem: Recall that µ p : W 1,p (Ω) → R is the C 1 -functional on W 1,p (Ω), defined by

On account of Propositions 3 and 4, together with H(ξ), H(β)
and H 0 , we have for some c 1 > 0 and all u ∈ W 1,p (Ω).
Then the nonlinear regularity theory of Lieberman [18] implies that From (9) we have for a.a. z ∈ Ω, thus u ∈ D + (see [30, pp. 111, 120]. Next we show the uniqueness of this positive solution. To this end we consider the integral functional k : if u ≥ 0 and u 1/q ∈ W 1,p (Ω), and k(u) = +∞ otherwise. From Diaz-Saa [8,

Lemma 1] and H(ξ) and H(β), k is convex and by Fatou's Lemma it is also lower semicontinuous.
Suppose v λ is another solution of (Au λ ). For this solution too we have v λ ∈ D + .
Then for h ∈ C 1 (Ω) and all |t| < 1 small we have v λ + th ∈ dom k and u λ + th ∈ dom k, where dom k = {u ∈ L 1 (Ω) : k(u) < +∞}, which is the effective domain of k. We can easily show that k is Gateaux differentiable at u q λ ∈ D + and at v q λ ∈ D + in the direction of h. Moreover, the Chain Rule and the nonlinear Green's identity (see [13, p.211] imply that The convexity of k implies the monotonicity of k . Hence we have This proves the uniqueness of the positive solution of problem (Au λ ).
Proof. Let u λ ∈ D + be the unique positive solution of (Au λ ) established in Proposition 9. Consider the following truncation of the reaction: This is Caratheodory. Set K λ (z, x) = x 0 k λ (z, s)ds and consider the C 1 -functional ϕ λ : W 1,p (Ω) → R defined by Also, let ψ λ ∈ C 1 (W 1,p (Ω)) be as in the proof of Proposition 10. From (18) and (23) we have From Proposition 10 and its proof we have that u 0 ∈ int C 1 (Ω) [ u λ , w] and u 0 is a minimizer of ψ λ .
From (28) and [24,Proposition 7] we have u ∈ L ∞ (Ω), thus the nonlinear regularity theory of [18] implies that u ∈ D + and hence This proves Claim 1. Now we assume that K ϕ λ is finite.
This proves Claim 2.
Consider a sequence {u n } ⊆ W 1,p (Ω) such that