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 $\lambda$ is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.

In this problem, ξ ∈ L s (Ω) with s > N and ξ(·) is indefinite (that is, sign-changing). We assume that ξ(·) is bounded from above (that is, ξ + ∈ L ∞ (Ω)). So, the differential operator (the left-hand side) of problem (P λ ) is not coercive. In the reaction (the right-hand side) of (P λ ), we have the parametric 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 ARcondition. So, problem (P λ ) can be viewed as a perturbation of the classical eigenvalue problem for the operator u → −∆u + 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 was by 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 used the ARcondition 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 have already mentioned earlier, there is no potential term in all the aforementioned works, 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 was on the existence and multiplicity of positive solutions. So, the conditions on the perturbation f (z, ·) were different, leading to a bifurcation-type result describing the change in the set of positive solutions as the parameter λ moves in • who also produced 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 (−∆u + ξu, H 1 (Ω)) with Robin boundary condition, then the problem has seven nontrivial smooth solutions and we also provide 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 We denote by · 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 int C + = {u ∈ C + : u(z) > 0 for all z ∈ Ω}.
The linear map γ 0 (·) is compact from In the sequel, for the sake of notational simplicity, we shall drop the use of the map γ 0 (·). All restrictions of Sobolev functions on ∂Ω will be understood in the sense of traces.
As we have already mentioned in the introduction, our analysis of problem (P λ ) relies on the spectrum of u → −∆u + 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).
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 that λ 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 have already mentioned,û k ∈ C 1 (Ω) for all k ∈ N. We denote by E(λ k ) 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".
It follows from (3) that the elements of E(λ 1 ) have fixed sign, while by (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). We denote byû 1 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 in general not 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, Papageorgiou, Rȃdulescu & Repovš [14,Chapter 5]).
, 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 space 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 Next, we denote by δ k,m the Kronecker symbol, that is, Finally, let 2 * denote the Sobolev critical exponent corresponding to 2, that is, Now we introduce the hypotheses on the perturbation f (z, x).

Remarks. Hypotheses
Hence the function f (z, ·) is superlinear for a.a. z ∈ Ω. 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 (see Mugnai [6]). Integrating (5a) and using (5b), we obtain the following weaker condition x for a.a. z ∈ Ω and all |x| M (see (5a)).
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 shall employ the less restrictive condition H 1 (iii), which allows the consideration of superlinear nonlinearities with "slower" growth near ±∞, which fail to satisfy the AR-condition. The following example illustrates this fact. For the sake of simplicity, we shall 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 shall 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 K ϕ + λ ⊆ C + (regularity theory), So, we may assume that K ϕ + λ is finite. Otherwise we already have an infinity of positive smooth solutions and 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) so small that Claim. The functional ϕ + λ satisfies the C-condition. Consider a sequence {u n } n 1 ⊆ H 1 (Ω) such that |ϕ + λ (u n )| C 2 for some C 2 > 0 and all n ∈ N, From (14) we have ε n for all n ∈ N, ε n for all n ∈ N (see (2)), ⇒ u − n → 0 in H 1 (Ω) as n → ∞.
We use (20) in (19) and conclude that (21) {u + n } n 1 ⊆ L τ (Ω) is bounded. 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 n r r C 6 u + n tr for some C 6 > 0 and all n ∈ N (23) (see (21) and use the Sobolev embedding theorem).
From hypothesis H 1 (i) we have for a.a. z ∈ Ω, all x 0 and some C 7 > 0.
So, we choose η > r big enough so that tr < 2 and reasoning as above, we obtain (26) and invoking the Kadec-Klee property, we again reach (29). We conclude that ϕ + λ satisfies the C-condition. This proves the claim.
It follows from (11) and (30) thatû = u 0 . If we show thatû = 0, then this will be the second positive solution of (P λ ).
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 solutions v 0 ,v ∈ −int C + , v 0 =v.

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.
Since 2 < r, we can find ρ ∈ (0, 1) so small that The proof is now complete.
Therefore we can conclude that The proof is now complete. Now we are ready to produce the seventh nontrivial smooth solution of problem (P λ ).
The proof is now complete.
So, summarizing our findings for problem (P λ ), we can state the following multiplicity theorem.
Open problem. Is it possible to show thatỹ is nodal (see [3,8])? Also, it seems that we cannot generate more than seven solutions without symmetry hypotheses (see [1]).