On the spectrum of a nontypical eigenvalue problem

We study a nontypical eigenvalue problem in a bounded domain from the Euclidian space R2 subject to the homogeneous Dirichlet boundary condition. We show that the spectrum of the problem contains two distinct intervals separated by an interval where there are no other eigenvalues.


The statement of the problem
Let Ω ⊂ R 2 be an open and bounded domain with smooth boundary denoted by ∂Ω.We consider the following problem where λ is a real parameter and h : R → R is the function given by with p ∈ (0, 1) a fixed real number.Note that this equation is not a typical eigenvalue problem since it has an inhomogeneous character (in the sense that if u is a nontrivial solution of the equation then tu fails to be its solution for all t ∈ R).However, since in this paper we are interested in finding parameters λ ∈ R for which problem (1.1) has nontrivial solutions we will call it a nontypical eigenvalue problem.In this context, we will call such a parameter Corresponding author.Email: mmihailes@yahoo.coman eigenvalue of problem (1.1) and a corresponding nontrivial solution of the equation an eigenfunction.Moreover, we will refer to the set of all eigenvalues of problem (1.1) as being the spectrum of the problem.To be more precise, we will use the following definition in our subsequent analysis.
Definition 1.1.We say that λ ∈ R is an eigenvalue of problem (1.1) if there exists u ∈ H 1 0 (Ω) \ {0} such that Function u from the above relation is called an eigenfunction associated to eigenvalue λ.

Background, motivation and main result
First, we recall that in the case when h(t) = t, for all t ∈ R, problem (1.1) reduces to the celebrated eigenvalue problem of the Laplace operator, i.e.
It is well-known that the spectrum of problem (1.4)In particular, we just recall that the first eigenvalue of problem (1.4) is obtained by minimizing the Rayleigh quotient associated to the problem Furthermore, each eigenfunction corresponding to λ 1 has constant sign in Ω.
On the other hand, in the case when the function h involved in problem (1.1) is of the form with f satisfying the properties (I) there exists a positive constant C ∈ (0, 1) such that | f (t)| ≤ Ct for any t ≥ 0; (II) there exists t 0 > 0 such that it was proved in [6, Theorem 1] that the spectrum of problem (1.1) contains, on the one hand, the isolated eigenvalue λ 1 given by relation (1.5) and, on the other hand, a continuous part, consisting in an interval (µ 1 , ∞) with µ 1 > λ 1 .Finally, we consider the case when h(t) = e t , for all t ∈ R. Then problem (1.1) reads as follows −∆u(x) = λe u(x) , x ∈ Ω, Problems of type (1.7) have been extensively studied in the literature (see, e.g.[2] or [3] and the reference therein).For instance in [2, Theorem 1.3 & Theorem 5.8] it was proved that there exist two positive constants µ 1 and µ 2 (with µ 1 < µ 2 ) such that each λ ∈ (0, µ 1 ) is an eigenvalue of problem (1.7) while any λ ∈ (µ 2 , ∞) can not be an eigenvalue of problem (1.7).
Motivated by the above results, in this paper we study equation (1.1) when function h involved in its formulation is given by relation (1.2).We reveal a new situation which can occur in the description of the spectrum of this problem, namely the fact that it contains two separate intervals.More precisely, we prove the following result.Theorem 1.2.Assume function h from problem (1.1) is given by relation (1.2) and λ 1 is given by relation (1.5).Then there exist two positive real numbers λ and λ with λ < λ such that each λ ∈ (0, λ ) ∪ (λ , ∞) is an eigenvalue of problem (1.1).Moreover, any λ ∈ ( λ 1 2 , λ 1 ) is not an eigenvalue of problem (1.1).

Proof of the main result
In order to prove Theorem 1.2 we start by recalling a series of known results that will be essential in the analysis of problem (1.1).Chapter 8] for more details) we can define the Orlicz space
A well-known result (see, e.g.[8, pp. 221-222]) asserts that the Sobolev space H 1 0 (Ω) is continuously embedded in the Orlicz space L Φ 0 (Ω), where Φ 0 (t) := e t 2 − 1, for all t ∈ R.This result is a consequence of Trudinger's inequality (see [9] or [4,Theorem 7.15]) which ensures that there exist two positive constants c 1 and c 2 (independent of Ω) such that Actually, the above inequality can be improved (see, e.g.[7]), since there exists a constant Finally, note that for any N-function Ψ that satisfies the property: lim t→∞ 2 , for all t ∈ R, we observe that and consequently Similarly, it can be checked that H 1 0 (Ω) is compactly embedded in each Lebesgue space L q (Ω), with q ∈ (1, ∞).

Proof of Theorem 1.2
First, we note that the embedding of H 1 0 (Ω) into the Orlicz spaces L Φ 0 (Ω), L Ψ 0 (Ω) and L p+1 (Ω) guarantees the fact that the integrals involved in Definition 1.1 are well-defined and, thus, problem (1.1) is well-posed.
The proof of Theorem 1.2 will be a simple consequence of the conclusions of Propositions 2.1, 2.3 and 2.5 below.
Proof.Let λ > 0 be an eigenvalue for problem (1.1) with its corresponding eigenfunction u ∈ H 1 0 (Ω) \ {0}.Note that since u = 0 then at least one of the functions u + and u − is nontrivial in Ω.
which, in view of the fact that 1 − e −y ≤ y for all y ≥ 0, yields − dx > 0 and the above facts imply λ ≥ λ 1 . (2.4) Otherwise, if u − ≡ 0 and λ > 0 is an eigenvalue for problem (1.1) then u + = 0 and relation (2.3) reads as follows Testing in the above relation with φ = u + we find which implies u + H 1 0 (Ω) = 0, or u + ≡ 0. Consequently, problem (2.8) possesses only nonpositive eigenfunctions.Thus, it is enough to analyse the problem In other words, conditions (H1) − (H3) from [6, page 320] are fulfilled with h(x, t) = h 1 (t).Similar arguments as those used in the proofs of [6,Lemmas 4 & 5] can be considered in order to show that following result.
Finally, we consider the problem Testing in the above relation with φ = u − we obtain We infer that u − H 1 0 (Ω) = 0 which implies that u − ≡ 0. Thus, problem (2.10) possesses only nonnegative eigenfunctions.Taking into account the definition of an eigenvalue for problem (1.1) (see relation (2.3)) we deduce that an eigenvalue of problem (2.10) is in fact an eigenvalue for problem (1.1).
In order to go further we introduce the Euler-Lagrange functional associated to problem (2.10), i.e.J λ : H 1 0 (Ω) → R defined by Standard arguments assure that J λ ∈ C 1 (H 1 0 (Ω); R) and its derivative is given by We note that the weak solutions of problem (2.10) are exactly the critical points of the functional J λ .In view of Definition 2.4, λ is an eigenvalue of problem (2.10) if and only if the functional J λ has a nontrivial and nonnegative critical point.
Remark 2.6.Since p ∈ (0, 1), the Hilbert space H 1 0 (Ω) is compactly embedded in the Lebesgue space L p+1 (Ω) with p + 1 ∈ (1, 2) which implies that there exists a positive constant C such that In order to prove Proposition 2.5 it is useful to first establish two auxiliary results.
Proof.Taking into account that e y − y − 1 ≥ 0, for all y ≥ 0, we deduce that for any t ∈ (0, 1) we have dx .
The proof of Lemma 2.8 is complete.

.14)
On the other hand, by Lemma 2.8 there exists t 1 > 0 sufficiently small such that J λ (t 1 e 1 ) < 0, where e 1 is a positive eigenfunction associated to λ 1 from relation (1.5).Moreover, taking into account relations (2.11) and (2.13) we deduce that It is clear to see that u is a minimum point of I λ and thus for small positive t and any u ∈ B 1 (0).The above relation yields Letting t → 0 we infer that J λ (u ), u + u H 1 0 (Ω) ≥ 0 and this implies that J λ (u ) ≤ .Thus, there exists a sequence {u n } ⊂ B 1/2 (0) such that J λ (u n ) → c and J λ (u n ) → 0, as n → ∞. (2.15) It is clear that sequence {u n } is bounded in H 1 0 (Ω) which implies that there exists u ∈ H 1 0 (Ω) such that, up to a subsequence, still denoted by {u n }, {u n } converges weakly to u in H 1 0 (Ω).which implies that u ∈ B 1/2 (0).By the compact embedding of H 1 0 (Ω) in L Ψ 0 (Ω) and L p+1 (Ω), we deduce that {u n } converges strongly to u in L Ψ 0 (Ω) and L p+1 (Ω).(2.17) Subtracting (2.16) from (2.17) and using the above pieces of information we deduce that Therefore, we obtain that {u n } converges strongly to u in H 1 0 (Ω), and using (2.15) we deduce that J λ (u) = c < 0 and J λ (u) = 0 .
We conclude that u is a nontrivial critical point of functional J λ .Since J λ (v) ≥ J λ (|v|) for any v ∈ H 1 0 (Ω), it follows that u is a nonnegative and nontrivial critical point of J λ .Thus, any λ ∈ (0, λ ) is an eigenvalue of problem (2.10).The proof of Proposition 2.5 is complete.