DOUBLE RESONANCE FOR ROBIN PROBLEMS WITH INDEFINITE AND UNBOUNDED POTENTIAL

. We study a semilinear Robin problem driven by the Laplacian plus an indeﬁnite and unbounded potential term. The nonlinearity f ( x,s ) is a Carath´eodory function which is asymptotically linear as s → ±∞ and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval (cid:104) ˆ λ k , ˆ λ k +1 (cid:105) . Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.


Introduction.
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω. In this paper, we study the following semilinear Robin problem −∆u + ξ(x)u = f (x, u) in Ω, where ξ : Ω → R is a potential function being in general indefinite (that is, sign changing) and unbounded (more precisely, ξ ∈ L q (Ω) for q > N ). The nonlinearity f : Ω × R → R is a Carathéodory function, that is, x → f (x, s) is measurable for all s ∈ R and s → f (x, s) is continuous for a.a. x ∈ Ω. In addition, in the boundary condition of (1.1), ∂u ∂n denotes the normal derivative defined by extension of the linear map u → ∂u ∂n = (∇u, n) R N for all u ∈ C 1 (Ω), with n : Ω → R being the outward unit normal on ∂Ω. The boundary coefficient β belongs to W 1,∞ (∂Ω) satisfying β(x) ≥ 0 for all x ∈ ∂Ω. When β = 0, we recover the Neumann problem.
In this paper we assume that f (x, ·) is asymptotically linear as s → ±∞ and resonant, that is, the quotient f (x,s) By H 1 (Ω) we denote the usual Hilbert space with inner product (u, h) = where · 2 stands for the norm in the Lebesgue space L 2 (Ω). The norm of R N is denoted by · R N and (·, ·) R N stands for the inner product in R N . For s ∈ R, we set s ± = max{±s, 0} and for u ∈ H 1 (Ω) we define u ± (·) = u(·) ± . It is well known that The Lebesgue measure on R N is denoted by | · | N and for a measurable function h : Ω × R → R (for example, a Carathéodory function), we define the Nemytskij operator corresponding to the function h by N h (u)(·) = h(·, u(·)) for all u ∈ H 1 (Ω).
Evidently, x → N h (u)(x) is measurable. In addition to the Sobolev space H 1 (Ω) we will also use the ordered Banach space C 1 (Ω) and its positive cone This cone has a nonempty interior in C(Ω) given by int C 1 (Ω) + = u ∈ C 1 (Ω) + : u(x) > 0 for all x ∈ Ω .
On the boundary ∂Ω we consider the (N − 1)-dimensional Hausdorff (surface) measure σ(·). Having this measure, we can define in the usual way the boundary Lebesgue spaces L s (∂Ω) for 1 ≤ s ≤ ∞. From the theory of Sobolev spaces we know that there exists a unique linear map γ 0 : H 1 (Ω) → L 2 (∂Ω) known as the trace map such that γ 0 (u) = u ∂Ω for all u ∈ H 1 (Ω) ∩ C(Ω).
The trace map prescribes boundary values to Sobolev functions. Furthermore we know that the trace map is compact into L s (∂Ω) for every s ∈ [1, 2 * ), where 2 * is the critical exponent of 2 given by Moreover it holds im γ 0 = H 1 2 ,2 (∂Ω) and ker γ 0 = H 1 0 (Ω). From now on, for the sake of notational simplicity, we drop the usage of the trace operator γ 0 . All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces.
As we already described in the Introduction we will use the spectrum of the differential operator u → −∆u + ξ(x)u with Robin boundary condition. So, we consider the following linear eigenvalue problem (2.1) In this eigenvalue problem we impose the following conditions on its data: From D'Aguì-Marano-Papageorgiou [6] we know that there exist µ, c 0 > 0 such that Then using (2.2) and the spectral theorem for compact self-adjoint operators on a Hilbert space, we can have a complete description of the spectrum of (2.1). This consists of a strictly increasing sequence λ k k∈N of eigenvalues such thatλ k → +∞ as k → +∞. By E λ k , k ∈ N, we denote the corresponding eigenspace. These are finite dimensional subspaces of H 1 (Ω) and we have the following orthogonal direct sum decomposition The eigenvalues of (2.1) have the following properties: The infimum in (2.3) is attained on E(λ 1 ) while both the infimum and the supremum in (2.4) are attained on E(λ m ). Evidently the elements of E(λ 1 ) have fixed sign while the elements of E(λ m ), m ≥ 2, are nodal, that is, sign changing. Byû 1 we denote the L 2 -normalized (that is, û 1 2 = 1) positive eigenfunction corresponding toλ 1 . If ξ ∈ L q (Ω) with q > N , then the regularity theory of Wang [18] (see Lemmata 5.1 and 5.2) implies thatû 1 ∈ C 1 (Ω) + \ {0}. Moreover, Harnack's inequality (see, for example, Motreanu-Motreanu-Papageorgiou [12, p. 212]) we haveû 1 (x) > 0 for all x ∈ Ω. If ξ + ∈ L ∞ (Ω), then the strong maximum principle (see, for example, p. 738]) implies thatû 1 ∈ int C 1 (Ω) + . If ξ ∈ L q (Ω) with q > N 2 , then the eigenspaces E(λ k ), k ∈ N have the so-called "Unique Continuation Property" (ucp for short) which says that if u ∈ E(λ k ) and u vanishes on a set of positive Lebesgue measure, then u ≡ 0.
The above properties lead to the following useful inequalities which can be found in D'Aguì-Marano-Papageorgiou [6]. Lemma 2.3.
Next, let us recall some basic definitions and facts about Morse theory which will need in the sequel. Let X be a Banach space and let (Y 1 , Y 2 ) be a topological pair such that Y 2 ⊆ Y 1 ⊆ X. For every integer k ≥ 0 the term H k (Y 1 , Y 2 ) stands for the k th =-relative singular homology group with integer coefficients. Recall that and so the quotient 328 NIKOLAOS S. PAPAGEORGIOU AND PATRICK WINKERT makes sense.
Given ϕ ∈ C 1 (X) and c ∈ R, we introduce the following sets: (the critical set of ϕ), K c ϕ = {u ∈ K ϕ : ϕ(u) = c} (the critical set of ϕ at the level c).
For every isolated critical point u ∈ K c ϕ the critical groups of ϕ at u ∈ K c ϕ are defined by The excision property of singular homology theory implies that the definition of critical groups above is independent of the particular choice of the neighborhood U .
Suppose that ϕ ∈ C 1 (X) satisfies the C-condition and that inf ϕ(K ϕ ) > −∞. Let c < inf ϕ(K ϕ ). The critical groups of ϕ at infinity are defined by This definition is independent of the choice of the level c < inf ϕ(K ϕ ). This is a consequence of the second deformation theorem (see, for example, Gasiński-Papageorgiou [7, p. 628]).
We now assume that K ϕ is finite and introduce the following series in t ∈ R: The Morse relation says that u∈Kϕ M (t, u) = P (t, ∞) + (1 + t)Q(t) for all t ∈ R, (2.6) with Q(t) = k≥0 β k t k being a formal series in t ∈ R with nonnegative integer coefficients. By A ∈ L(H 1 (Ω), H 1 (Ω) * ) we denote the linear operator defined by Furthermore, we say that a Banach space X has the Kadec-Klee property, if the following implication is true: Locally uniformly convex Banach spaces, in particular Hilbert spaces, have the Kadec-Klee property. Finally, let that is,λ m0 is the first nonnegative eigenvalue of (2.1). If ξ ≥ 0 and β ≥ 0, then λ 1 ≥ 0 and so m 0 = 1. Moreover, if ξ ≥ 0, β ≥ 0 and one of the two is different from zero, thenλ 1 > 0.
3. Multiplicity Theorem. In this section we prove a multiplicity theorem about the existence of three nontrivial solutions to problem (1.1) under conditions of double resonance. Our hypotheses on the data of problem (1.1) are the following. H(ξ): ξ ∈ L q (Ω) with q > N when N ≥ 2 and q = 1 when N = 1; in addition x ∈ Ω and for all s ∈ R; uniformly for a.a.
Now let µ > 0 as in (2.2) and consider the following truncation-perturbation of the nonlinearity f (x, ·) It is clear that both functions are of Carathéodory type. We setF ± (x, s) = s 0f ± (x, t)dt and introduce the C 1 -functionalsφ ± : Furthermore, let ϕ : H 1 (Ω) → R be the energy (Euler) functional of problem (1.1) which is defined by Evidently ϕ ∈ C 1 (H 1 (Ω)). Let us consider first the truncation functionalsφ ± .
Proof. We will show the assertion only forφ + , the proof forφ − is similar. So, let (u n ) n≥1 ⊆ H 1 (Ω) be a sequence such that We claim now that (u n ) n≥1 ⊆ H 1 (Ω) is bounded. Arguing by contradiction, suppose that by passing to a subsequence if necessary, we have u + n → ∞.

) results in
Hypotheses x ∈ Ω and for all s ∈ R with c 4 > 0.
Similarly we show that the functionalφ − satisfies the C-condition.
Let us now prove a similar result for the energy functional ϕ : Proof. Consider a sequence (u n ) n≥1 ⊆ H 1 (Ω) such that |ϕ(u n )| ≤ M 2 for some M 2 > 0 and for all n ≥ 1, (3.13) Assertion (3.14) implies On the other hand, by using (3.13), we have Adding (3.16) and (3.17) we obtain for some M 3 > 0 and for all n ∈ N. We are going to show that the sequence (u n ) n≥1 ⊆ H 1 (Ω) is bounded. Arguing indirectly, suppose that at least for a subsequence, we have u n → +∞.
Let y n = un un for all n ∈ N. Then y n = 1 for all n ∈ N and so we may assume that y n y in H 1 (Ω) and y n → y in L 2q q−1 (Ω) and in L 2 (∂Ω). (3.19) Inequality (3.15) can be rewritten as follows for all h ∈ H 1 (Ω). As before, see Papageorgiou-Rȃdulescu [14], this is equivalent to Hence, as before, u n → u in H 1 (Ω). Therefore we conclude that ϕ satisfies the C-condition. Proof. As before we are going to show the assertion only for the functionalφ + , the proofs forφ − and for ϕ are very similar.
That means u = 0 is a strict local minimizer ofφ + . Similarly we show that u = 0 is a strict local minimizer forφ − and ϕ.
So, by the Sobolev embedding theorem we have W 2,q (Ω) → C 1,α (Ω) with α = 1 − N q > 0. Therefore  Proof. Based on Proposition 3.4 we see that we can assume that the critical sets Kφ + and Kφ − are finite or otherwise we already have an infinity of positive and negative solutions of problem (1.1) and so we are done. Proposition 3.3 implies the existence of ρ ∈ (0, 1) small enough such that because of k ≥ 2 andû 1 ∈ int C 1 (Ω) + . Finally, Proposition 3.1 stateŝ ϕ + satisfies the C-condition.

NIKOLAOS S. PAPAGEORGIOU AND PATRICK WINKERT
Similarly, working with the functionalφ − instead, we prove the existence of a negative solution v 0 ∈ − int C 1 (Ω) + .
In order to prove the existence of a third smooth solution of problem (1.1) we will apply Morse theory in terms of critical groups.
Proof. Let λ ∈ λ k ,λ k+1 and consider the C 2 -functional ψ : H 1 (Ω) → R defined by The choice of λ > 0 implies that K ψ = {0} and u = 0 is nondegenerate. Hence we have for all t ∈ [0, 1] and for all u ∈ H 1 (Ω). Claim. We can find τ ∈ R and ς > 0 such that for all t ∈ [0, 1] and for all u ∈ H 1 (Ω). Suppose that the Claim is not true. Since h is bounded on bounded sets we can find (t n ) n≥1 ⊆ [0, 1] and (u n ) n≥1 ⊆ H 1 (Ω) such that t n → t, u n → +∞, h(t n , u n ) → −∞ and From the last convergence in (3.39) we have for all h ∈ H 1 (Ω) with ε n → 0 + . Let y n = un un for all n ≥ 1. Then y n = 1 for all n ≥ 1 and so we may assume that y n y in H 1 (Ω) and y n → y in L Passing to the limit in (3.42) as n → ∞ and applying (3.43), we obtain Reasoning as in the proof of Proposition 3.2 by applying (3.44), (3.45), (3.46) and hypothesis H(f )(iii), we reach a contradiction. So, (3.39) cannot be true, hence the Claim holds. As in the proof of Proposition 3.1 we can easily show that h(t, ·) satisfies the C-condition for every t ∈ [0, 1]. Therefore, Proposition 3.2 of Liang-Su [11] (see also Chang [5]) gives Becuase of (3.38) we derive We also compute the critical groups at infinity of the functionalsφ ± . Recall that without any loss of generality, we can assume that the critical sets Kφ ± are finite. Proof. We will do the proof only forφ + , the proof forφ − is very similar.

Claim 1.
We can findτ ∈ R andς > 0 such that Again we argue by contradiction. So suppose that we can find (t n ) n≥1 ⊆ [0, 1] and (u n ) n≥1 ⊆ H 1 (Ω) such that t n → t, u n → +∞,ĥ + (t n , u n ) → −∞ and see the proof of Proposition 3.6. The last convergence in (3.47) gives for all n ∈ N with ε n → 0 + . Recall that Passing to the limit in (3.51) as n → ∞ and applying (3.52) we obtain Now, by applying (3.53) and (3.54) and following the ideas in the proof of Proposition 3.1, we reach a contradiction and this proves Claim 1.
The homotopy invariance of the singular homology groups implies that for r > 0 small enough, we obtain  In a similar way we show that Now we can have an exact computation of the critical groups of the energy functional ϕ at the two constant sign solutions u 0 ∈ int C 1 (Ω) + as well as v 0 ∈ − int C 1 (Ω) + produced in Proposition 3.5. Note the fact that ϕ is not C 2 (recall that f is only a Carathédory function) does not permit the usage of classical results from Morse theory (see, for example, Motreanu-Motreanu-Papageorgiou [12]) and makes the computation of the critical groups of ϕ at u 0 ∈ int C 1 (Ω) + and at v 0 ∈ − int C 1 (Ω) + a nontrivial, interesting task.
Therefore, (u n ) n≥1 is a sequence of distinct positive solutions of (1.1), a contradiction. Hence, (3.69) cannot occur and so from the homotopy invariance of critical groups (see, for example, Gasiński-Papageorgiou [9, p. 836]) we have Because of y 0 ∈ K ϕ , we know that y 0 is a third nontrivial solution of (1.1) and the regularity theory of Wang [18] implies that y 0 ∈ C 1 (Ω).