Superlinear Perturbations of the Eigenvalue Problem for the Robin Laplacian Plus an Indefinite and Unbounded Potential

We consider a superlinear perturbation of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential. Using variational tools and critical groups, we show that when λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.

(P λ ) linear term u → λu and a perturbation f (z, x) which is a measurable function such that f (z, ·) is continuously differentiable. We assume that f (z, ·) exhibits superlinear growth near ±∞, but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short). Instead we employ a less restrictive condition that incorporates in our framework superlinear nonlinearities with slower growth near ±∞ which fail to satisfy the AR-condition. So, problem (P λ ) can be viewed as a perturbation of the classical eigenvalue problem for the operator u → −Δu+ξ(z)u with Robin boundary condition.
In the past, such problems were studied primarily in the context of Dirichlet equations with no potential term. The first work is that of Mugnai [5], who used a general linking theorem of Marino & Saccon [4] to produce three nontrivial solutions. The work of Mugnai was extended by Rabinowitz, Su & Wang [18] who based their method of proof on bifurcation theory, variational techniques and critical groups in order to produce three nontrivial solutions. Analogous results for scalar periodic equations were proved by Su & Zeng [20]. All the aforementioned works use the AR-condition to express the superlinearity of the perturbation f (z, ·). A more general superlinearity condition was employed by Ou & Li [7] who also produced three nontrivial solutions for λ > 0 near a nonprincipal eigenvalue. As we already mentioned earlier, in all the aforementioned works, there is no potential term and so the differential operator is coercive. This facilitates the analysis of the problem. Papageorgiou, Rȃdulescu & Repovš [13] went beyond Dirichlet problems and studied Robin problems with an indefinite potential. In [13] the emphasis is on the existence and multiplicity of positive solutions. So, the conditions on the perturbation f (z, ·) are different, leading to a bifurcation-type result describing the change in the set of positive solutions as the parameter λ moves in • R + = (0, +∞). We also mention the works of Castro, Cassio & Velez [1], Papageorgiou & Papalini [8] (Dirichlet problems), and Hu & Papageorgiou [3] (Robin problems) who also produce seven nontrivial solutions. In Castro, Cassio & Velez [1] there is no potential term, while Papageorgiou & Papalini [8] and Hu & Papageorgiou [3] have an indefinite potential term and moreover, provide sign information for all solution they produce. For related results we refer to Papageorgiou & Rȃdulescu [10], Papageorgiou & Winkert [16], Papageorgiou & Zhang [17], and Rolando [19]. Finally, we mention the work of Papageorgiou & Rȃdulescu [12] who proved multiplicity results for nearly resonant Robin problems.
In the present paper, using variational tools from the critical point theory together with suitable truncation, perturbation and comparison techniques and using also critical groups (Morse theory), we show that when the parameter λ > 0 is close to an eigenvalue of (−Δu + ξu, H 1 (Ω)) with Robin boundary condition, then the problem has seven nontrivial smooth solutions, also providing sign information for six of them.

Mathematical Background and Hypotheses
The main spaces in the analysis of problem (P λ ) are the Sobolev space H 1 (Ω), the Banach space C 1 (Ω) and the "boundary" Lebesgue spaces L p (∂Ω), 1 p ∞.
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. So The Banach space C 1 (Ω) is ordered by the positive (order) cone This cone has a nonempty interior given by On ∂Ω we consider the (N − 1)-dimensional Hausdorff measure (surface measure) σ(·). Using this measure, we can define in the usual way the boundary value spaces L p (∂Ω), where 1 p ∞. 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", such that So, the trace map extends the notion of boundary values to all Sobolev functions. We know that im γ 0 = H 1/2,2 (∂Ω) and ker γ 0 = H 1 0 (Ω). The linear map γ 0 (·) is compact from H 1 (Ω) into L p (∂Ω) for all p ∈ 1, 2(N −1) In the sequel, for the sake of notational simplicity, we drop the use of the map γ 0 (·). All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces.
As we mentioned in the introduction, our analysis of problem (P λ ) relies on the spectrum of u → −Δu + ξ(z)u with Robin boundary condition. So, we consider the following linear eigenvalue problem We say thatλ ∈ R is an "eigenvalue", if problem (1) admits a nontrivial solutionû ∈ H 1 (Ω) known as an "eigenfunction" corresponding to the eigenvalueλ. From hypotheses H 0 and the regularity theory of Wang [21], we know thatû ∈ C 1 (Ω).
Let γ : From D'Agui, Marano & Papageorgiou [2] (see also Papageorgiou & Rȃdulescu [11]), we know that there exists μ > 0 such that Using (2) and the spectral theorem for compact, self-adjoint operators on a Hilbert space, we show (see [2,11]) that the spectrum of (1) consists of a sequence {λ k } k∈N of distinct eigenvalues such thatλ k → +∞ as k → ∞. There is also a corresponding sequence {û k } k∈N ⊆ H 1 (Ω) of eigenfunctions which form an orthogonal basis for H 1 (Ω) and an orthonormal basis for L 2 (Ω). As we already mentioned,û k ∈ C 1 (Ω) for all k ∈ N. By E(λ k ) we denote the eigenspace corresponding to the eigenvalueλ k . We have E(λ k ) ⊆ C 1 (Ω) for all k ∈ N, this subspace is finite dimensional and Moreover, each eigenspace E(λ k ) has the "unique continuation property" (the UCP for short) which says that "if u ∈ E(λ k ) and u(·) vanishes on a set of positive measure, then u ≡ 0 .
The first (principal) eigenvalueλ 1 is simple, that is, dim E(λ 1 ) = 1. All the eigenvalues admit variational characterizations in terms of the Rayleigh quotient γ(u) In (3) the infimum is realized on E(λ 1 ), while in (4) both the supremum and the infimum are realized on E(λ k ).
From (3) it follows that the elements of E(λ 1 ) have fixed sign, while from (4) and the orthogonality of the eigenspaces, we see that the elements of E(λ k ) (for k 2) are nodal (that is, sign-changing). Byû 1 we denote the positive, L 2 -normalized (that is, û 2 = 1) eigenfunction corresponding toλ 1 . The regularity theory and the Hopf maximum principle imply thatû 1 ∈ int C + .
Let X be a Banach space, c ∈ R and ϕ ∈ C 1 (X, R). We introduce the following sets We say that ϕ(·) satisfies the "C-condition", if the following property holds: "Every sequence {u n } n 1 such that {ϕ(u n )} n 1 ⊆ R is bounded and (1 + u n X )ϕ (u n ) → 0 in X * as n → ∞, admits a strongly convergent subsequence . This is a compactness-type condition on the functional ϕ(·). Since the ambient space is not in general locally compact (being infinite dimensional), the burden of compactness is passed to the functional ϕ(·). Using the C-condition one can prove a deformation theorem from which follows the minimax theory of the critical values of ϕ(·) (see, for example, then the critical groups of ϕ at u are defined by The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the isolating neighborhood U . We say that a Banach X has the "Kadec-Klee property" if the following is true A uniformly convex space has the Kadec-Klee property. In particular, Hilbert spaces have the Kadec-Klee property. We denote by A ∈ L(H 1 (Ω), H 1 (Ω) * ) the operator defined by Also, by δ k,m we denote the Kronecker symbol defined by Finally, let 2 * denote the Sobolev critical exponent corresponding to 2, that is, Now we introduce the hypotheses on the perturbation f (z, x).
(vii) for every ρ > 0, there existsξ ρ > 0 such that for a.a. z ∈ Ω, the function Hence for a.a. z ∈ Ω, the function f (z, ·) is superlinear. However, this superlinearity of the perturbation term is not expressed using the AR-condition, which is common in the literature when dealing with superlinear problems. Recall that the AR-condition says that there exist q > 2 and M > 0 such that
So we see that the AR-condition implies that f (z, ·) has at least (q − 1)polynomial growth. In this paper, instead of the AR-condition, we employ the less restrictive condition H 1 (iii), which allows the consideration of superlinear nonlinearities with "slower" growth near ±∞, which fail to satisfy the ARcondition. The following example illustrates this fact. For the sake of simplicity, we drop the z-dependence of f and assume that ξ ∈ L ∞ (Ω). Suppose that for some m ∈ N, we have C |λ m+2 | + ξ ∞ , C > 0. Then the function satisfies hypotheses H 1 but fails to satisfy the AR-condition. For all λ > 0, let ϕ λ : H 1 (Ω) → R denote the energy functional associated to problem (P λ ), which is defined by We have ϕ λ ∈ C 2 (H 1 (Ω)).

Constant Sign Solutions
In this section we prove the existence of four nontrivial smooth constant sign solutions when λ ∈ (λ m ,λ m+1 ).

From (5) it is clear that
It is easy to see that So, we may assume that K ϕ + λ is finite. Otherwise we already have an infinity of positive smooth solutions and so we are done. Then on account of Theorem 5.7.6 of Papageorgiou, Rȃdulescu & Repovš [14, p. 449], we can find ρ 0 ∈ (0, 1) small such that Hypothesis H 1 (ii) implies that Claim. The functional ϕ + λ satisfies the C-condition. Consider a sequence {u n } n 1 ⊆ H 1 (Ω) such that From (14) we have In (15) ε n for all n ∈ N, ε n for all n ∈ N (see (2)), Next, we choose h = u + n ∈ H 1 (Ω) in (15). We obtain On the other hand from (13) and (16), we have We add (17) and (18) and obtain Hypotheses H 1 (i), (iii) imply that we can findβ 1 ∈ (0,β 0 ) and C 5 > 0 such thatβ We use (20) in (19) and obtain that First assume that N 3. From hypothesis H 1 (iii) we see that without any loss of generality, we may assume that τ < r < 2 * . So, we can find t ∈ (0, 1) such that tr for some C 6 > 0, all n ∈ N (see (21) and use the Sobolev embedding theorem).
So, we choose η > r big enough so that tr < 2 and reasoning as above, we obtain (26) and then from that and the Kadec-Klee property, we reach again (29). We conclude that ϕ + λ satisfies the C-condition. This proves the Claim. Then (11), (12) and the Claim, permit the use of the mountain pass theorem. So, we can findû ∈ H 1 (Ω) such that (11)).
From (11) and (30) it follows thatû = u 0 . If we show thatû = 0, then this will be the second positive solution of (P λ ).
On account of hypotheses H 1 (i), (iv), we have We have C 11 u 2 + u r for some C 11 > 0 (see (31)). (32) Also for h ∈ H 1 (Ω) we have By hypothesis, λ ∈ (λ m ,λ m+1 ) and m 2. So, u = 0 is a nondegenerate critical point of ϕ λ with Morse index d m = dim H m 2 (since m 2). Then by Proposition 6.2.6 of [14, p. 479], we have On the other hand, from the previous part of the proof we know that u ∈ K ϕ + λ is of mountain pass type. Therefore Theorem 6.5.8 of Papageorgiou, Rȃdulescu & Repovš [14, p. 527] implies that From (36), (35) and since d m 2, we conclude thatû = 0 and soû ∈ int C + is the second positive solution of (P λ ) distinct from u 0 .
For the negative solutions, we consider the Carathéodory function g − λ (z , x) defined by Working with these two functionals as above, we produce two negative

Nodal Solutions
In this section we show that when λ is close toλ m+1 (near resonance) we can generate two nodal (sign-changing) solutions.

The Seventh Nontrivial Solution
In this section we prove the existence of a seventh nontrivial solution for problem (P λ ) when λ ∈ (λ m ,λ m+1 ). However, we are unable to provide sign information for this seventh solution.