Existence of positive eigenfunctions to an anisotropic elliptic operator via the sub-supersolution method

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Introduction.
In this paper, the main goal is to show the existence of positive solutions of the problem where Ω ⊂ R N , N ≥ 1, is a bounded and regular domain, p i > 1, i = 1, . . . , N, q > 1, and λ is a real parameter. We will assume without loss of generality that the p i are ordered increasingly, that is, p 1 ≤ · · · ≤ p N . There is a vast literature concerning anisotropic elliptic problems. We mention here only those references most strongly related to (1.1). First, in [9], it was proved that for q < p N and for any γ > 0, there exist λ γ > 0 and a solution u γ of (1.1) with u γ p = γ and λ = λ γ . As the authors themselves claim, from this result, the existence of solutions of (1.1) can not be deduced for a given λ. In [4], using mainly variational methods, it was proved that if p 1 < q < p N , then there exist 0 < λ * ≤ λ * such that: • If λ ≤ λ * , (1.1) does not possess a positive solution.
In this paper, we complete and improve the above results. For that, we use the sub-supersolution method, see [1,5], and [16] (see also [6][7][8], and references therein for the application of this method to problems with nonlinear reaction functions including singularities or critical exponent). The existence and regularity theory involving p-Laplacian type equations is nowadays mature enough to allow the discovery of new results by application to special cases suitably chosen. This method allows us not only to prove the existence of a solution, but also gives us lower and upper bounds of such a solution. Roughly speaking, we construct by hand sub-and supersolutions to the main equation, inspired by the usual way of determining energy estimates for these kind of equations. Specifically, our main result is the following. An outline of the paper is the following: in Section 2, we recall some definitions and some properties of the eigenvalues and eigenfunctions of the classical p-Laplacian. Next, in Section 3, we enunciate the sub-supersolution method. Then in Section 4, we construct sub-and supersolutions by multiplication of powers of p-Laplacian eigenfunctions to be applied in the existence theorem.

Preliminary lemmas and setting.
Consider h(x, s) : Ω × R → R a Caratheodory function, i.e., measurable in x and continuous in the second variable s. Consider the anisotropic problem (2.1) The natural framework to study (2.1) is the anisotropic Sobolev space W 1,p 0 (Ω), that is, the closure of C ∞ 0 (Ω) under the anisotropic norm then for every r ∈ [1, p ∞ ], the embedding W 1,p 0 (Ω) ⊂ L r (Ω) is continuous, and it is compact if r < p ∞ . More precisely, it holds the following directional Poincaré-type inequality for any u ∈ C 1 c (Ω) (see for instance [9]) where {e 1 , . . . , e N } denotes the canonical basis of R N . The theory of embeddings of this kind of anisotropic Sobolev spaces is vast and we refer to [9] for the directional Poincaré-type inequality and to [12] for Sobolev and Morrey's embeddings of the whole W 1,p (Ω) space, obtained with an important geometric condition on the domain Ω, namely that it must be semi-rectangular. We briefly recall the definition for later use.
It is not the case that this semi-rectangular condition reflects in our construction of the solution: the existence of traces for this kind of functions is heavily depending on the geometry of the domain as shown in [12]. Regularity theory for anisotropic operators like the one defined by equation (1.1) is still a challenging open problem, see for example [3]. We also recall the following definition of a weak solution.
Now we introduce some well-known results concerning the eigenvalue problem for the p-Laplacian, that will be of crucial importance for the construction. Specifically, we consider the problem The following result is well-known: Lemma 2.1. The eigenvalue problem (2.6) has a unique eigenvalue λ = λ 1 with the property of having a positive associated eigenfunction ϕ 1 ∈ W 1,p 0 (Ω), called principal eigenfunction. Moreover, λ 1 is simple, isolated, and is defined by Furthermore, ϕ 1 ∈ C 1,β (Ω) for some β ∈ (0, 1) and ∂ϕ 1 /∂n < 0 on ∂Ω, where n is the outward unit normal on ∂Ω. Finally, for N = 1, we have that and in fact Remark 2.1. The existence of λ 1 and the main properties of ϕ 1 are well-known, see [10,13,15]. Property (2.7) holds for N = 1, see for instance [11], and for N ≥ 2 in some specific domains, for example, for Ω convex, see [2].
3. An existence sub-supersolution theorem. We start by stating an important theorem, for whose proof, we refer to [1, Theorem 5.1], that assures the existence of a solution between a sub-and a supersolution. subsolution and supersolution of (2.1) such that u ≤ u. Then there exist a solution u ∈ W 1,p 0 (Ω) of (2.1) such that u ≤ u ≤ u.
Proof. By continuity, the function h verifies condition (h 2 ) of [1]. This concludes the proof.

Construction of sub-and supersolutions: proof of the main result.
In this section, we prove Theorem 1.1. For that, we apply Theorem 3.1 to (1.1). Mainly, we construct the sub-and the supersolutions.
As every v i is bounded, it is clear that u ∈ W 1,p (Ω) and that u |∂Ω = 0. Hence, u is a subsolution of (1.1) provided that On the other hand, observe that Then, taking into account the positivity of v i , ∀ i = 1, . . . , N, with the obvious notation for dx i . Next, by using an integration by parts argument and the Fubini-Tonelli theorem, we get The second term on the right can be discarded as ∂vi ∂ni < 0 in ∂U i , see (4.2). Considering that ∂xi and, by construction, that our subsolution condition is implied by the following inequality (4.5) Let us require a condition on the pointwise integrand
On the other hand, if we take S 0 under consideration, we observe that the behaviour next to ∂U is controlled: when v i → 0 + , then, as ∂ ∂ni v i < 0 on ∂U i , we have that there exists δ > 0 small enough such that the quantity Then S 0 is bounded or negative in U and we can conclude that λ * is finite.

Supersolutions.
Since Ω is bounded, we can choose a domain U such that Now for M > 0, we consider the function where v i are the first eigenfunctions to the p i -Laplacian in U i , whose first eigenvalue we denote by η i . Observe that Then u is a supersolution of (1.1) if for all 0 ≤ φ ∈ W 1,p 0 (Ω) it holds It is clear that Thus we may ask for the strong condition Hence, if 1 < q < p N , by letting M → ∞, we have that u is a supersolution ∀λ > 0.
Proof of Theorem 1.1. If λ ≤ 0, the maximum principle assures that there does not exist a positive solution of (1.1). 1. Assume 1 < q < p 1 . Fix λ > 0. Then we can choose > 0 small and M large enough such that u, u are sub-supersolutions of (1.1) and u ≤ u in Ω. Theorem 3.1 assures the existence of a solution u of (1.1) such that u ≤ u ≤ u. This completes this case. 2. Assume p 1 ≤ q < p N . In this case, taking for example = 1, we have that u is a subsolution provided that λ ≥ λ * for some λ * . On the other hand, we can take M large such that u is supersolution and u ≤ u. Thus, there exists a positive solution for λ ≥ λ * . Now, we define Λ := inf{λ : (1.1) possesses at least one positive solution}.
We have proved that Λ < ∞. For p 1 < q < p N , in [4], it was proved that 0 < Λ. We show now that this is also true for q = p 1 . Indeed, let us now consider q = p 1 and let us multiply the equation ( We prove now that for all λ > Λ, we have the existence of a positive solution. Indeed, fix λ 0 > Λ. Then, by definition of Λ, there exists μ ∈ (Λ, λ 0 ) and a positive solution, denoted by u μ , of (1.1) for λ = μ. Since μ < λ 0 , it is clear that u μ is a subsolution of (1.1) for λ = λ 0 . On the other hand, for M large, there exists a supersolution u of (1.1) for λ = λ 0 . Finally, thanks to regularity results, see for instance [1,Proposition 4.1], we have that u μ ∈ L ∞ (Ω). Hence, for M large, u μ ≤ u, and we can can be faced with the same technique, using properties of the p i -Laplacian principal eigenfuctions and the integrability condition (2.7) recently obtained in [2].