POSITIVE AND NODAL SOLUTIONS FOR PARAMETRIC NONLINEAR ROBIN PROBLEMS WITH INDEFINITE POTENTIAL

. We consider a parametric nonlinear Robin problem driven by the p − Laplacian plus an indeﬁnite potential and a Carath´eodory reaction which is ( p − 1) − superlinear without satisfying the Ambrosetti - Rabinowitz condition. We prove a bifurcation-type result describing the dependence of the set of positive solutions on the parameter. We also prove the existence of nodal solutions. Our proofs use tools from critical point theory, Morse theory and suitable truncation techniques.

The potential function ξ ∈ L ∞ (Ω) is indefinite (that is, sign changing) and λ > 0 is a parameter. The reaction term f (z, x) is a Carathéodory function (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous) which is (p − 1)− superlinear in the x−variable, but without satisfying the usual (in such cases) Ambrosetti -Rabinowitz condition. Indeed, we replace such a condition with a weaker one which lets us consider superlinear nonlinearities with slower growth near ∞, and not satisfying the Ambrosetti -Rabinowitz condition. Finally, in the boundary condition, ∂u ∂n p denotes the generalized normal derivative defined by ∂u ∂n p = |Du| p−2 (Du, n) R N for all u ∈ W 1,p (Ω), with n(·) being the outward unit normal on ∂Ω. This kind of normal derivative is dictated by the nonlinear Green's identity (see, for example, Gasinski -Papageorgiou [11, p. 211]) and in an even more general form can be found also in the work of Lieberman [15]. The boundary weight function β(·) belongs to C 0,α (∂Ω) with α ∈ (0, 1) and β(z) ≥ 0 for all z ∈ ∂Ω. When β ≡ 0, we have the Neumann problem and in that case from (1) we see that the boundary condition becomes ∂u ∂n = 0 on ∂Ω (the usual normal derivative). First, we look for positive solutions and our aim is to establish the precise dependence of the set of positive solutions on the parameter λ > 0. So, we prove a bifurcation-type result and show that there exists a critical parameter value λ * > 0 such that for all λ > λ * problem (P λ ) admits at least two positive solutions, for all λ = λ * problem (P λ ) admits at least one positive solution, for all λ < λ * problem (P λ ) has no positive solutions.
We also show that for every λ ∈ [λ * , +∞) problem (P λ ) has a smallest positive solution u * λ and we investigate the monotonicity and continuity properties of the map λ → u * λ in the relevant function space. Finally, in Section 4, we impose bilateral asymptotic conditions on f (z, ·) and prove the existence of nodal (sign changing) solutions.
Our approach is variational, based on the critical point theory. In Section 4, in order to generate a nodal solution, we also use tools from Morse theory (critical groups).
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, R), we say that ϕ satisfies the Cerami condition (the "C-condition" for short), if the following property holds: admits a strongly convergent subsequence. This is a compactness-type condition on functional ϕ which compensates for the fact that the ambient space X need not be locally compact (X is in general infinite dimensional). The C-condition is a basic tool in probing a deformation theorem for the sublevel sets of ϕ, from which one can deduce the minimax theory for the critical values of ϕ. Prominent in that theory is the well-known "mountain pass theorem" due to Ambrosetti -Rabinowitz [3]. Here we state it in a slightly more general form (see, for example, Gasinski -Papageorgiou [11, p. 648]).
then c ≥ m ρ and c is a critical value of ϕ.
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) and the boundary space L τ (∂Ω) with τ ∈ [1, +∞].
In what follows by | · | we denote the norm on R N , by (·, ·) R N the inner product of R N and by · the norm of the Sobolev space W 1,p (Ω) defined by The space C 1 (Ω) is an ordered Banach space with positive cone This cone has a nonempty interior given by On ∂Ω we consider the (N − 1)− dimensional Hausdorff (surface) measure σ(·). With this measure on ∂Ω, we can define the Lebesgue spaces L τ (∂Ω), 1 ≤ τ ≤ ∞. From the theory of Sobolev spaces, we know that there exists a unique linear continuous map γ 0 : W 1,p (Ω) → L p (∂Ω), known as the "trace map", s.t. γ 0 (u) = u| ∂Ω for all u ∈ W 1,p (Ω) ∩ C(Ω). So, we understand the trace map as representing the "boundary values" of a Sobolev function u ∈ W 1,p (Ω). The trace map is compact into L r (∂Ω) for all r ∈ 1, N p − p N − p when p < N and into L r (∂Ω) for all r ∈ [1, +∞) when p ≥ N . We know that In the rest of the work, for the sake of notational simplicity, we drop the use of the trace map γ 0 . It is understood that all restrictions of Sobolev functions u ∈ W 1,p (Ω) on ∂Ω are defined in the sense of traces.

Remark 1.
In fact the result is still true even when f 0 (z, ·) has critical growth, namely r = p * , see Papageorgiou -Radulescu [25].
We will also need a strong comparison result which is of independent interest and for this reason is formulated in a more general setting.
Proof. To fix things we assume that ∂v ∂n a ∂Ω < 0.
As we already mentioned in the Introduction, in Section 4 in order to produce a nodal solution, we will use tools from Morse theory (critical groups). So, let us recall the definition of critical groups at an isolated critical point.
Let X be a Banach space, ϕ ∈ C 1 (X, R) and c ∈ R. We introduce the following sets: we denote the k th relative singular homology group with integer coefficients. If u ∈ K c ϕ is isolated, then the critical groups of ϕ at u are defined by The excision property of singular homology theory implies that this definition of critical groups is independent of the particular choice of the neighborhood U as above. Finally, we fix our notation. By | · | N we denote the Lebesgue measure on R N .
3. Positive solutions. In order to look for positive solutions, our hypotheses on the data of problem (P λ ) are the following: x µ uniformly for a.a.z ∈ Ω; (iv) there exists δ > 0 and q ∈ (1, p) such that Remark 4. The alternative in H(β) means that we exclude mixed problems.
Remark 5. Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis R + = [0, +∞), without any loss of generality, we may assume that f (z, x) = 0 for a.a. z ∈ Ω and all x ≤ 0. Hypotheses We point out that we do not employ the Ambrosetti -Rabinowitz condition, which says that there exist τ > p and M > 0 such that essinf Ω F (·, M ) > 0 (see Ambrosetti -Rabinowitz [3] and Mugnai [19]).
By direct integration of the Ambrosetti -Rabinowitz condition, we havê Hence, F (z, ·) has at least τ − polynomial growth near +∞.
Example. The following functions satisfy hypotheses H(f ) (for the sake of simplicity, we drop the z− dependence): Note that f 2 does not satisfy the Ambrosetti -Rabinowitz condition.
Let L = {λ > 0 : problem (P λ ) has a positive solution} (this is the set of admissible parameters) and S(λ) the set of positive solutions of problem (P λ ) (for λ ∈ L, we have S(λ) = ∅).

Remark 6.
In fact a careful reading of the above proof, reveals that the solutions of (P λ ) exhibit the following "monotonicity" property. Given λ ∈ L and θ > λ, then θ ∈ L and there exists u θ ∈ S(θ) ⊆ int C + such that u θ ≤ u λ . This property can be improved provided we strengthen the Hypotheses on f (z, ·).
Proof. As we already mentioned in the previous Remark, we have: θ ∈ L and we can find u θ ∈ S(θ) ⊆ int C + s.t.u θ ≤ u λ .
Proof. We argue indirectly. So, suppose that λ * = 0 and let λ n ↓ 0 as n → ∞. Then {λ n } n≥1 ⊆ L (see Proposition 5). Moreover, from the last part of the proof of Proposition 5, we know that we can find {u n } n≥1 ⊆ W 1,p (Ω) such that u n ∈ S(λ n ) ⊆ int C + (see Proposition 4) and ϕ λn (u n ) < 0 for all n ∈ N.
Hypotheses H(f ) 2 (i), (iii) imply that we can find γ ∈ (0, γ 0 ) and c 1 > 0 such that Using this estimate in (30) we obtain tat It is clear from Hypothesis H(f ) 2 (iii), that we may assume, without loss of generality, that µ < r.
From (32) we get so from (35) we infer that . So, for the argument above to work, it is enough to replace p * by s > r large enough. More precisely, from (32) we have and so for s > r big, we have tr < p. Then the previous argument works and again we have that {u n } n≥1 ⊆ W 1,p (Ω) is bounded. So, we may assume that u n u * in W 1,p (Ω) and u n → u * in L p (Ω) and in L p (∂Ω) as n → ∞.
We will show thatũ λ ∈ int C + is the unique positive solution of (P) λ . Indeed, suppose thatṽ λ is another positive solution of (P) λ . As above, we can show that v λ ∈ int C + . otherwise.
Then, recalling that µ > ξ ∞ and the definition ofβ and ofĝ, we get that for some M 2 > 0, all n ∈ N.
In (64) we choose h = u + n ∈ W 1,p (Ω). Then for all n ∈ N. Adding (63) and (65) we obtain for some M 3 > 0 and all n ∈ N. Hence, by (53) and (54), we immediately find that for some M 4 > 0 and all n ∈ N. Using (66) and reasoning as in the proof of Proposition 7 (see the part of the proof from (30) until (37)), we deduce that there exists a subsequence of {u n } n≥1 ⊆ W 1,p (Ω) such that u n → u in W 1,p (Ω). This proves the claim.
Next we examine what happens at the critical parameter value λ * > 0 (see Proposition 7). Proof. Let {λ n } n≥1 ⊆ (λ * , ∞) be such that λ n ↓ λ * . We can find u n ∈ S(λ n ) ⊆ int C + , n ∈ N, such that ϕ λn (u n ) < 0 for all n ∈ N (see the proofs of Propositions 5 and 7). Then, reasoning as in the proof of Proposition 7, we show that {u n } n≥1 ⊂ W 1,p (Ω) is bounded and so we may assume that u n u * in W 1,p (Ω) and u n → u * in L r (Ω) and in L p (∂Ω). for all h ∈ W 1,p (Ω) and every n ∈ N. If in (69) we choose h = u n − u * ∈ W 1,p (Ω), pass to the limit as n → ∞ and use (68), then lim n→∞ A(u n ), u n − u * = 0, so that, by Proposition 1, u n → u * in W 1,p (Ω) as n → ∞.
Therefore, if we pass to the limit as n → ∞ in (69), then we have for all h ∈ W 1,p (Ω). Hence u * is a solution of problem (P λ * ), u * ∈ C + by nonlinear regularity. From the proof of Proposition 7 we know that u λ1 ≤ u n for all n ∈ N (ũ λ1 ∈ int C + ), and soũ λ1 ≤ u * . As a consequence, u * ∈ S(λ * ) ⊆ int C + and λ * ∈ L.
Next we show the existence of a smallest positive solution for problem (P λ ), λ ∈ L.
Evidently {u n } n≥1 ⊆ W 1,p (Ω) is bounded and so we mat assume that u n u * λ in W 1,p (Ω) and u n → u * λ in L p (Ω) and in L p (∂Ω).
Summarizing the situation for the positive solutions of problem (P λ ), λ > 0, we have the following multiplicity theorem (bifurcation -type result).