Resonant Anisotropic (p,q)-Equations

We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

In (1), the left hand side (the differential operator) is the sum of two such operators with different exponents. Equations driven by the sum of two differential operators of different nature, such as the anisotropic (p, q)-equations of the present work, arise in the mathematical models of many physical processes. We refer to the survey papers of Marano-Mosconi [1] and Rȃdulescu [2] and the references therein. In particular, anisotropic equations arise in elasticity (see Zhikov [3,4]) and in the study of electrorheological and magnetorheological fluids (see Rǔžička [5] and Versaci-Palumbo [6]). For other papers dealing with the sum of two differential operators of different nature (mostly (p, q)-Laplacian) we refer to Candito-Gasiński-Livrea [7], Gasiński-Klimczak-Papageorgiou [8], , Gasiński-Winkert [13,14], and for anisotropic problems governed by the p(z)-Laplacian we refer to Gasiński-Papageorgiou [15,16]. Finally for the use of the eigenproblem to molecules we refer to Jäntsch [17], and Teng-Lu [18].
In the reaction (right hand side of (1)), the function 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 asymptotically as x → ±∞, we can have resonance with respect to the principal eigenvalue of (−∆ p(z) , W 1,p(z) 0 (Ω)). Using variational tools from the critical point theory, together with suitable truncation techniques and Morse theory (critical groups), we show that problem (1) has at least three nontrivial smooth solutions, one positive, one negative and the third nodal (sign-changing). For isotropic problems, such three solutions theorem was proved for Dirichlet problems driven by the p-Laplacian by Liu [19] (Theorem 1.2). In that paper, the reaction f (z, ·) asymptotically as x → ±∞ is uniformly nonresonant with respect to the principal eigenvalue of (−∆ p , W 1,p 0 (Ω)) and no nodal solutions are obtained. For the same problem, the resonant case was examined by Liu-Su [20], who obtained two nontrivial solutions, but without providing sign information for them.
The study of anisotropic equations is lagging behind and there is only the work of Fan-Zhao [21], who produced nodal solutions for a class of radially symmetric equations driven by the anisotropic p-Laplacian. Our work here appears to be the first one producing three nontrivial smooth solutions with sign information for resonant anisotropic (p, q)-equations.

Mathematical Background-Hypotheses
The study of problem (1) requires the use of Lebesgue and Sobolev spaces with variable exponents. For a comprehensive presentation of such spaces, we refer to the book of Diening-Harjulehto-Hästö-Rǔžička [22].
By M(Ω) we denote the space of all functions u : Ω → R which are measurable. As usual, we identify two such functions which differ only on a Lebesgue-null set. Given r ∈ E 1 , the variable exponent Lebesgue space L r(z) (Ω) is defined by We equip this space with the so-called "Luxemburg norm", defined by The space (L r(z) (Ω), · r(z) ) is separable and uniformly convex (hence reflexive too, by the Milman-Pettis theorem; see Papageorgiou-Winkert [23] (Theorem 3.4.28, p. 225) or Gasiński-Papageorgiou [24] (Theorem 5.89, p. 853)). If then r ∈ E 1 and we have L r(z) (Ω) * = L r (z) (Ω). In addition the following Hölder-type inequality holds These function spaces have many properties similar to the classical Lebesgue L p -space. So, if r 1 , r 2 ∈ E 1 and r 1 (z) r 2 (z) for all z ∈ Ω, then L r 2 (z) (Ω) embeds continuously into L r 1 (z) (Ω).
Using the variable exponent Lebesgue spaces, given r ∈ E 1 , we can define the variable exponent Sobolev space W 1,r(z) (Ω) as follows In this definition, the gradient Du is understand in the weak sense. The space W 1,r(z) (Ω) is equipped with the following norm For the sake of notational simplicity, we will write Du r(z) = |Du| r(z) . When r ∈ E 1 ∩ C 0,1 (Ω) (that is, r ∈ E 1 is Lipschitz continuous on Ω), then we can also define The spaces W 1,r(z) (Ω) and W 1,r(z) 0 (Ω) are separable and uniformly convex (thus reflexive). For the space W 1,r(z) 0 (Ω), the well-known Poincaré inequality is still valid, namely there exists c > 0 such that This inequality implies that on W 1,r(z) 0 (Ω) we can consider the equivalent norm (Ω).
Then the following embeddings are true The study of the anisotropic Lebesgue and Sobolev spaces uses the following modular function for r ∈ E 1 . This modular function is closely related to the Luxemburg norm.

Proposition 2.
The operator A r(z) is bounded (that is, maps bounded sets into bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type (S) + , which means that (Ω) and lim sup (Ω)." In addition to the anisotropic spaces, we will also use the space This is an ordered Banach space with positive (order) cone This cone has a nonempty interior given by with n being the outward unit normal on ∂Ω.
In contrast to the isotropic case, in the anisotropic case it can happen that inf L = 0 (see Fan-Zhang-Zao [25] (Theorem 3.1)). However, if there exists η ∈ R N (N > 1) such that for all z ∈ Ω, the function ϑ(t) = r(z + tη) is monotone on T z = {t ∈ R : z + tη ∈ Ω} and if r ∈ C 1 (Ω), then there exists a principal eigenvalue λ 1 (r) > 0 with corresponding eigenfunction u 1 (r) ∈ int C + (see Fan-Zhang-Zao [25] (Theorem 3.3)). Moreover, we have Let X be a Banach space, ϕ ∈ C 1 (X) and c ∈ R. We introduce the following sets we denote the k-th relative singular homology group for the pair (Y 1 , Y 2 ) with integer coefficients. Let u ∈ K ϕ be isolated and ϕ(u) = c. Then the critical groups of ϕ at u are defined by The excision property of singular homology implies that this definition of critical groups is independent of the choice of the neighbourhood U.
We say that a set S ⊆ W 1,p(z) 0 (Ω) is "downward directed" if, for every pair u 1 , u 2 ∈ S, we can find u ∈ S such that u u 1 , u u 2 ; similarly, we say that S is "upward directed", if for every pair v 1 , v 2 ∈ S, we can find v ∈ S such that v 1 v, v 2 v. Finally as for the Luxemburg norm, we write (Ω). Now we are ready to introduce the hypotheses on the data of problem (1). Hypothesis 1. p ∈ C 1 (Ω) and there exists η ∈ R N such that for all z ∈ Ω, the function t −→ ϑ(t) = p(z + tη) is monotone on T z = {t ∈ R : z + tη ∈ Ω}, q ∈ E 1 and q(z) < p(z) for all z ∈ Ω.

Example 1.
The following function satisfies hypotheses H 1 : Figure 1. We will need the following lemma.

Lemma 1.
If ϑ ∈ L ∞ (Ω), ϑ(z) λ 1 for a.a. z ∈ Ω and ϑ ≡ λ 1 , then there exists c 1 > 0 such that Proof. We proceed indirectly. So, suppose that the lemma is not true. Then we can find a sequence (Ω) such that Evidently we may assume that u n 0 for all n ∈ N. We have Suppose that the sequence {u n } n∈N ⊆ W 1,p(z) 0 (Ω) is not bounded. Then by passing to a subsequence if necessary we may assume that u n −→ ∞, so, by Proposition 1, also We set y n = u n (Ω), n ∈ N. Differentiating, we have so where | · | N denotes the Lebesgue measure on R N . Then Proposition 1 and the Poincaré inequality, imply that the sequence Passing to a suitable subsequence if necessary, we may assume that (Ω) and y n −→ y in L p(z) (Ω).
From (10) and reasoning as above (replacing y n with u n ), we reach again a contradiction. So, the assertion of the lemma is true.

Solutions of Constant Sign
We introduce the energy (Euler) functional for problem (1) and the positive and negative truncations of it. So, we consider the following three functionals ϕ, ϕ ± : W 1,p(z) 0 (Ω) −→ R: (Ω).
Proof. We do the proof for the functional ϕ + , the proof for the functionals ϕ − and ϕ being similar.
In a similar fashion, we show that the functionals ϕ − and ϕ are coercive too. Now that we have the coercivity of the functionals ϕ ± , we can use the direct method of the calculus of variations to produce two constant sign solutions.

Proposition 4.
If hypotheses H 0 and H 1 hold, then problem (1) has at least two constant sign solutions u 0 ∈ int C + and v 0 ∈ −int C + , both local minimizers of the energy functional ϕ.
Note that ϕ| C + = ϕ + | C + . Therefore we see that u 0 is a local C 1 0 (Ω)-minimizer of ϕ, so by Proposition 3.3 of Gasiński-Papageorgiou [15] and Theorem 3.2 of Tan-Fang [29], Similarly, working with the functional ϕ − , we produce a negative solution v 0 ∈ −int C + which is a local minimizer of ϕ.
In fact we can show that there exist extremal constant sign solutions, that is, a smallest positive solution and a biggest negative solution. In Section 4, we will use these extremal constant sign solutions in order to generate a nodal (sign-changing) solution.
To obtain the extremal constant sign solutions, we need to do some preliminary work.
Let S + (resp. S − ) be the set of positive (resp. negative) solutions of Problem (1). From Proposition 4 and its proof, we know that We will produce a lower bound for the set S + and an upper bound for the set S − . To this end, note that on account of hypotheses H 1 (i), (iv), we have for some c 8 > 0. This unilateral growth condition on f (z, ·) leads to the consideration of the following auxiliary anisotropic Dirichlet problem For this problem, we have the following existence and uniqueness result.

Proposition 5.
If hypotheses H 0 hold, then problem (23) has a unique positive solution u ∈ int C + . Moreover, since problem is odd v = −u ∈ −int C + is the unique negative solution of (23).
Next we show the uniqueness of this positive solution. For this purpose, we consider the integral functional j : From Theorem 2.2 of Takáč-Giacomoni [32], we know that j is convex. Suppose that y ∈ W 1,p(z) 0 (Ω) is another positive solution of (23). Again we have that y ∈ int C + . Then using Proposition 4.1.22 of Papageorgiou-Rȃdulescu-Repovš [33] (p. 274), we infer that (Ω), then for |t| < 1 small we have Hence the convexity of j implies the Gâteaux differentiability of j at u τ + and at y τ + in the direction h. Moreover, using the chain rule and Green's identity, we obtain The convexity of j implies the monotonicity of j . Therefore so u = y. This proves the uniqueness of the positive solution u ∈ int C + of (23). Since problem (23) is odd, it follows that v = −u ∈ −int C + is the unique negative solution of problem (23).
These solutions will serve as bounds of S + and S − respectively.

Proposition 6.
If hypotheses H 0 and H 1 hold, then u u for all u ∈ S + and v v for all v ∈ S − .

Proof.
We do the proof for the set S + , the proof for the set S − being similar. So, let u ∈ S + ⊆ int C + and introduce the Carathéodory function k + defined by We set and consider the C 1 -functional σ + : W 1,p(z) 0 (Ω) −→ R defined by (Ω). Evidently σ + is coercive (see (26)) and sequentially weakly lower semicontinuous. Therefore we can find u 0 ∈ W 1,p(z) 0 (Ω) such that If w ∈ int C + , then we can find t ∈ (0, 1) small such that tw u (recall that u ∈ int C + and use Proposition 4.1.22 of Papageorgiou-Rȃdulescu-Repovš [33] (p. 274)). Using (26) and the fact that τ + < q − < p − , we see that by taking t ∈ (0, 1) even smaller if necessary, we will have (Ω).
In a similar fashion, we show that v v for all v ∈ S − .
Next following some ideas of Filippakis-Papageorgiou [34], we show that S + is downward directed and S − is upward directed.

Proposition 7.
If hypotheses H 0 and H 1 hold, then S + is downward directed and S − is upward directed.
Proof. We do the proof for S + , the proof for S − being similar.
Similarly we show that S − is upward directed.
Then (36) and (37) imply that u * ∈ S + , u * = inf S + . Similarly working with the set S − , we produce v * ∈ S − such that v * = sup S − . Note that since S − is upward directed, we can find an increasing sequence {v n } n∈N such that sup n∈N v n = sup S − .

Nodal Solution
In this section we produce a nodal solution for problem (1). The idea is to use truncations in order to focus on the order interval Then on account of the extremality of the solutions u * and v * , any nontrivial solution of (1) located in [v * , u * ] and distinct from u * and v * will be nodal. To produce such a solution, we will combine tools from critical point theory and from Morse theory (critical groups).
We start with a result which provides the critical groups of the energy functional ϕ at the origin. The result is a consequence of Hypothesis H2(iv) and follows from Proposition 6 of Leonardi-Papageorgiou [37].
As mentioned above, to concentrate on the order interval [v * , u * ], we will use truncations. For this purpose, we introduce the function g defined by This is a Carathéodory function. We will also use the positive and negative truncations of g(z, ·), namely the Carathéodory functions We set G(z, x) = x 0 g(z, s) ds and G ± (z, x) = x 0 g ± (z, s) ds and consider the C 1 -functionals ξ, ξ ± : W 1,p(z) 0 (Ω) −→ R defined by for all u ∈ W 1,p(z) 0 (Ω).
Since u * ∈ int C + and v * ∈ −int C + , using Proposition 9 and a simple homotopy invariance argument as in the proof of Proposition 4.4 of Papageorgiou-Rȃdulescu-Repovš [38], we obtain the following result.

Proposition 10.
If hypotheses H 0 and H 1 hold, then C k ( ξ, 0) = C k (ϕ, 0) = 0 ∀k ∈ N 0 . Now we are ready to produce nodal solutions. Proposition 11. If hypotheses H 0 and H 1 hold, then problem (1) has a nodal solution Proof. Using (38) and (39), we can easily see that On account of the extremality of u * and v * , we have Claim. u * ∈ int C + and v * ∈ −int C + are local minimizers of ξ.
Similarly for v * ∈ −int C + using this time the functional ξ − . This proves the Claim.
Without any loss of generality, we may assume that The reasoning is similar if the opposite inequality holds. From (40) we see that we may assume that K ξ is finite. Otherwise we already have an infinity of nodal solutions of (1) and so we are done. By Theorem 5.7.6 of Papageorgiou-Rȃdulescu-Repovš [33] (p. 449), we can find ∈ (0, 1) small such that From (38) it is clear that ξ is coercive. So, using Proposition 5.1.15 of Papageorgiou-Rȃdulescu-Repovš [33] (p. 369), we have that ξ satisfies the Palais-Smale condition. (43) Then (42) and (43) permit the use of the mountain pass theorem. So, we can find y 0 ∈ W 1,p(z) 0 (Ω) such that (see (40) and (42)). From (42) and (44), it follows that y 0 ∈ {v * , u * }.
Finally we can state the following multiplicity theorem for Problem (1).

Remark 2.
In this paper we examined resonant anisotropic problems in which the resonance occurs from the left of λ 1 (see Hypothesis H2(ii)). This made the relevant energy functionals coercive (see Proposition 3). It is an interesting open problem what can be said if the resonance if from the right of λ 1 . In this case the functionals fail to be coercive.