Resonant semilinear Robin problems with a general potential

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term is a Carath\'eodory function which is resonant with respect to any nonprincipal eigenvalue both at $\pm \infty$ and 0. Using a variant of the reduction method, we show that the problem has at least two nontrivial smooth solutions.


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: In this problem, the potential function ξ(·) is unbounded and indefinite (that is, signchanging). So, in problem (1.1) the differential operator (on the left-hand side of the equation), is not coercive. The reaction term f (z, x) is a Carathéodory function (that is, for all x ∈ R, z → f (z, x) is measurable and for almost all z ∈ Ω, x → f (z, x) is continuous) and f (z, ·) exhibits linear growth as x → ±∞. In fact, we can have resonance with respect to any nonprincipal eigenvalue of −∆ + ξ(z)I with the Robin boundary condition. This general structure of the reaction term, makes the use of variational methods problematic. To overcome these difficulties, we develop a variant of the so-called "reduction method", originally due to Amann [1] and Castro & Lazer [3]. However, in contrast to the aforementioned works, the particular features of our problem lead to a reduction on an infinite dimensional subspace and this is a source of additional technical difficulties. In the boundary condition, ∂u ∂n is the normal derivative defined by extension of the continuous linear map u → ∂u ∂n = (Du, n) R N for all u ∈ C 1 (Ω), with n(·) being the outward unit normal on ∂Ω. The boundary coefficient β ∈ W 1,∞ (∂Ω) satisfies β(z) ≥ 0 for all z ∈ ∂Ω. We can have β ≡ 0, which corresponds to the Neumann problem.
Recently there have been existence and multiplicity results for semilinear elliptic problems with general potential. We mention the works of Hu & Papageorgiou [9], Kyritsi & Papageorgiou [10], Papageorgiou & Papalini [12], Qin, Tang & Zhang [17] (Dirichlet problems), Gasinski & Papageorgiou [6], Papageorgiou & Rȃdulescu [13,15] (Neumann problems) and for Robin problems there are the works of Shi & Li [18] (superlinear reaction), D'Agui, Marano & Papageorgiou [4] (asymmetric reaction), Hu & Papageorgiou (logistic reaction) and Papageorgiou & Rȃdulescu [16] (reaction with zeros). In all the aforementioned works the conditions are in many respects more restrictive or different and consequently the mathematical tools are different. It seems that our work here is the first to use this variant of the reduction method on Robin problems.

Mathematical background
Let X be a Banach space and let X * be 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 the functional ϕ and is more general that the usual Palais-Smale condition. The two notions are equivalent when ϕ is bounded below (see Motreanu,Motreanu & Papageorgiou [11,p. 104]).
Our multiplicity result will use the following abstract "local linking" theorem of Brezis & Nirenberg [2].
(that is, ϕ has a local linking at u = 0 with respect to the direct sum Y ⊕ V). Then ϕ has at least two nontrivial critical points.
The Sobolev space H 1 (Ω) is a Hilbert space with the following inner product By || · || we denote the norm corresponding to this inner product, that is, On ∂Ω we consider the (N − 1)-dimensional Hausdorff (surface) measure denoted by σ(·). Using this measure on ∂Ω, we can define in the usual way the Lebesgue spaces L r (∂Ω), 1 ≤ r ≤ ∞. From the theory of Sobolev spaces we know that there exists a unique continuous linear map γ 0 : H 1 (Ω) → L 2 (∂Ω), known as the "trace map", which satisfies So, the trace map assigns "boundary values" to any Sobolev function (not just to the regular ones). This map is compact into L r (∂Ω) for all r ∈ 1, 2(N−1) In what follows, for the sake of notational simplicity, we shall drop the use the trace map γ 0 . The restrictions of all Sobolev functions on ∂Ω, are understood in the sense of traces.
Next, we recall some basic facts about the spectrum of the differential operator −∆ + ξ(z)I with the Robin boundary condition. So, we consider the following linear eigenvalue problem: Our conditions on the data of (2.1) are the following: • ξ ∈ L s (Ω) with s > N; and Let γ : H 1 (Ω) → R be the C 1 -functional defined by By D'Agui, Marano & Papageorgiou [4], we know that there exists µ > 0 such that Using (2.2) and the spectral theorem for compact self-adjoint operators on a Hilbert space, we produce the spectrum σ 0 (ξ) of (2.1) and we have that σ 0 (ξ) = {λ k } k≥1 a sequence of distinct eigenvalues withλ k → +∞ as k → +∞. By E(λ k ) (for all k ∈ N), we denote the eigenspace corresponding to the eigenvalueλ k . We know that E(λ k ) is finite dimensional. Moreover, the regularity theory of Wang [19] implies that E(λ k ) ⊆ C 1 (Ω) for all k ∈ N. The Sobolev space H 1 (Ω) admits the following orthogonal direct sum decomposition The elements of σ 0 (ξ) have the following properties: The infimum in (2.3) is realized on E(λ 1 ), while both the infimum and supremum in (2.4) are realized on E(λ m ). It follows that the elements of E(λ 1 ) have fixed sign, while those of E(λ m ) (m ≥ 2) are nodal (sign-changing). The eigenspaces 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. As a consequence of the UCP, we have the following useful inequalities (see D'Agui, Marano & Papageorgiou [4]).
We have the following orthogonal direct sum decomposition So, every u ∈ H 1 (Ω) admits a unique sum decomposition of the form Finally, let us fix our notation. By | · | N we denote the Lebesgue measure on R N and by (by ·, · we denote the duality brackets for the pair (H 1 (Ω) * , H 1 (Ω))). Also, given a measurable function f : Ω × R → R (for example a Carathéodory function), we set (the critical set of ϕ).

Pair of nontrivial solutions
The hypotheses on the data of (1.1) are the following: Remark 3.1. We can have β ≡ 0 and this case corresponds to the Neumann problem.
Let ϕ : H 1 (Ω) → R be the energy (Euler) functional for problem (1.1) defined by The next proposition is crucial in the implementation of the reduction method.

Proposition 3.2. If hypotheses H(ξ), H(β)
, H( f ) hold, then there exists a continuous map τ : Proof. We fix v ∈ V and consider the C 1 -functional ϕ v : From the chain rule, we haveφ with p H * − being the orthogonal projection of the Hilbert space H 1 (Ω) onto H * − . By ·, · H − we denote the duality brackets for the pair (H This implies that −φ ′ v is strongly monotone and therefore −φ v is strictly convex.
From (3.10) and (3.11) it follows that Similarly we show that From (3.12) and (3.13) we conclude that
On the other hand from (3.8) we have . So, it follows from (3.14) and (3.15) that

From (3.26) via Fatou's lemma (hypothesis H( f )(iii) permits its use), we have
Using hypothesis H( f )(iii) we see that we can find M 2 > 0 such that Then We add (3.30) and (3.31) and obtain dz ≤ M 4 for some M 4 > 0, and all n ∈ N. (3.32) Comparing (3.29) and (3.32), we get a contradiction. This proves that {v n } n≥1 ⊆ V is bounded. So, we may assume that v n w → u in H 1 (Ω) and v n → u in L 2 (Ω) and L 2 (∂Ω). (3.33) In (3.19) we choose h = v n − u ∈ H 1 (Ω), pass to the limit as n → ∞ and use (3.33). Then ⇒ v n → u in H 1 (Ω) (as before via the Kadec-Klee property).
This proves Claim 3.6.
From Proposition 3.4, we deduce that: