A Multiplicity Theorem for Superlinear Double Phase Problems

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at +∞ and −∞ also negative.


Introduction
Let Ω ⊆ R N be a bounded domain (that is, a bounded connected set in the Ndimensional Euclidean space) with a smooth boundary ∂Ω. In this paper, we study the following double phase problem (P) −∆ a p u(z) − ∆ q u(z) = f (z, u(z)) in Ω, u| ∂Ω = 0, 1 < q < p < +∞ (see [1]), with the unknown u : Ω −→ R. If a ∈ L ∞ (Ω) \ {0}, a(z) 0 for almost all z ∈ Ω and 1 < r < +∞, then by ∆ a r we denote the weighted r-Laplacian differential operator defined by ∆ a r u = div (a(z)|Du| r−2 Du). If a ≡ 1, then we have the usual r-Laplacian. In problem (P) we have the sum of two such operators. So, the differential operator (left hand side) in (P) is not homogeneous. The differential operator of (P) is related to the so called "double phase" integral functional defined by J(u) = Ω a(z)|Du| p + |Du| q dz.
The integrand of this functional is the function ϑ(z, y) = a(z)|y| p + |y| q ∀(z, y) ∈ Ω × R N . (1) We do not assume that the weight a is bounded away from zero (that is, we do not assume that ess inf Ω a > 0). So, the function ϑ(z, ·) exhibits unbalanced growth, namely we have |y| q ϑ(z, x) c 0 |y| p + |y| q for a.a. z ∈ Ω, all y ∈ R N ,

Mathematical Background-The Nehari Manifold
As we already mentioned in the Introduction, the appropriate functional framework for the analysis of double phase problems, is provided by the so called Musielak-Orlicz spaces. For a comprehensive presentation of the theory of these spaces, we refer to the book of Harjulehto-Hästö [23].
We introduce the following hypotheses (denoted by H1) on the weight a and the exponents q, p, which will be used in the sequel. Hypothesis 1. a ∈ C 0,1 (Ω) \ {0} (that is, a : Ω −→ R is nonzero and Lipschitz continuous), a(z) 0 for all z ∈ Ω, 1 < q < p < N and p q < 1 + 1 N .

Remark 1.
The last condition on the exponents q, p is common in Dirichlet double phase problems and guarantees that the Poincaré inequality is valid for the corresponding Musielak-Orlicz-Sobolev space (see , p. 138)). Note that the inequality p q < 1 + 1 N implies that p < q * and as we will see later in this section, this guarantees useful compact embeddings among the relevant spaces.
As usual we identify two such functions which differ only on a Lebesgue-null subset of Ω.
Here and in the sequel Du denotes the weak gradient of u ∈ L ϑ (Ω). We equip W 1,ϑ (Ω) with the norm with Du ϑ = |Du| ϑ . We set For this space the Poincaré inequality holds (see , p. 138)), that is, there exists c > 0 such that u ϑ c Du ϑ ∀u ∈ W 1,ϑ 0 (Ω).
There is a close relation between the norm · and the modular function ϑ .
Let V : W 1,ϑ 0 (Ω) → W 1,ϑ 0 (Ω) * be the nonlinear map defined by for all u, h ∈ W 1,ϑ 0 (Ω) (by ·, · we denote the duality brackets for the pair (W 1,ϑ 0 (Ω), W 1,ϑ 0 (Ω) * )). This operator has the following properties (see , Proposition 3.1)). Proposition 3. The operator V : W 1,ϑ 0 (Ω) −→ W 1,ϑ 0 (Ω) * is bounded (that is, maps bounded sets into bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type (S) + , that is, V has the following property: "if u n w −→ u in W 1,ϑ 0 (Ω) and In the sequel by a we denote the modular function defined by For every x ∈ R, let Then for every u ∈ M(Ω), we set If u ∈ W 1,ϑ 0 (Ω), we know that By λ 1 (q) we denote the principal eigenvalue of (−∆ q , W 1,q 0 (Ω)), that is, we consider the nonlinear eigenvalue problem This problem has a smallest eigenvalue λ 1 (q) > 0, which has the following variational characterization This eigenvalue is simple and isolated and the corresponding eigenfunctions have fixed sign and belong in C 1 (Ω). In fact this is the only eigenvalue with eigenfunctions of fixed sign. All the other eigenvalues have eigenfunctions which are nodal. Moreover, if C + is the positive cone of for all z ∈ Ω} and u ∈ C 1 0 (Ω) is a positive eigenfunction corresponding to λ 1 (q) > 0, then with n being the outward unit normal on ∂Ω and ∂u ∂n = (Du, n) R N .
The next lemma is an easy consequence of the above properties (see for example , Lemma 4.11)).
Next we introduce our hypotheses on the reaction f (z, x).
Hypothesis 2. f : Ω × R −→ R is a Carathéodory function such that f (z, 0) = 0 for almost all z ∈ Ω and |x| p−2 x = +∞ uniformly for almost all z ∈ Ω; (iii) for almost all z ∈ Ω, the quotient function Remark 2. Hypothesis H2 (ii) implies that for almost all z ∈ Ω, f (z, ·) is (p − 1)-superlinear. However, we do not assume the usual in superlinear problems Ambrosetti-Rabinowitz condition (see, for example, Willem ([14], p. 46)). This condition is used by Liu-Dai [9] (see hypothesis ( f 4 ) in [9]). The Ambrosetti-Rabinowitz condition is a convenient but restrictive condition which permits the verification of the Palais-Smale condition (the compactness type condition on the energy functional), which is necessary in order to apply the minimax theorem of the critical point theory. Furthermore, hypothesis H2 (iii) is a relaxed version of Nehari-type monotonicity condition which requires that the quotient function x −→ f (z,x) |x| p−1 is strictly increasing on R \ {0}. This stronger version is used by Liu-Dai [9] (see hypothesis ( f 5 ) in [9]). Finally hypothesis H2 (iv) describes the behaviour of f (z, ·) near zero and is less restrictive than the corresponding condition in Liu-Dai [9] (see hypothesis ( f 3 ) in [9]), who requires that f (z, ·) is strictly (q − 1)-sublinear near zero. Hypotheses H2 are also less restrictive than those of Gasiński-Papageorgiou [8] (see hypotheses H( f ) in [8]). Example 1. The following functions satisfy hypotheses H2, but fail to satisfy the hypotheses of Gasiński-Papageorgiou [8] and Liu-Dai [9]: In these examples for the sake of simplicity we have dropped the z-dependence.
For η > 0, we consider the following perturbation of the reaction Note that f η satisfies the same hypotheses as f and in fact now we have that for almost all |x| p−1 is strictly increasing on R \ {0}. We consider the Dirichlet double phase problem with f η (z, x) as reaction. So, we consider the following problem First we will prove a multiplicity result for problem (P η ) and then let η → 0 + to have the multiplicity theorem for the original problem (P).
We set and consider the energy (Euler) functional ϕ η : Evidently ϕ η ∈ C 1 (W 1,ϑ 0 (Ω)) and we have Since we want to provide sign information for all the solutions produced, we introduce also the positive and negative truncations ϕ ± η : Again we have ϕ ± η ∈ C 1 (W 1,ϑ 0 (Ω)). We introduce the Nehari manifold N η for the functional ϕ η defined by Evidently the Nehari manifold N η contains the nontrivial weak solutions of problem (P η ). In order to produce constant sign solutions, we introduce the corresponding sets for the functionals ϕ ± η , namely the sets Finally for the purpose of producing nodal solutions, we introduce also the set Proposition 4. If hypotheses H1, H2 hold and u ∈ W 1,ϑ 0 (Ω) \ {0}, then there exists unique t u > 0 such that t u u ∈ N η .

Proof. Consider the function
Using (4) in (3), we have Since q < p, we see that there exists t 0 ∈ (0, 1) such that On the other hand, we have Using hypothesis H2 (ii) we see that Then (5), (6) and Bolzano's theorem, imply that there exists t u > 0 such that and observe that in the last equation the right hand side is strictly increasing in t > 0. Therefore, t u > 0 is unique.
The Nehari manifold is much smaller than the ambient space W 1,ϑ 0 (Ω). So, ϕ η | N η exhibits properties which fail to be true globally. In fact on account of hypothesis H2 (ii), ϕ η is unbounded below. However as we will show in the next proposition ϕ η | N η is coercive, hence bounded below. This illustrates the usefulness of the Nehari manifold.

Proof.
We argue by contradiction. So, suppose that we can find a sequence {u n } n∈N ⊆ N η such that ϕ η (u n ) c 1 for all n ∈ N and u n → +∞, for some c 1 > 0. Let y n = u n u n for n ∈ N. Then y n = 1 for all n ∈ N and so we may assume that (see Proposition 2). First we assume that y = 0. Let Then | Ω| N > 0, where by | · | N we denote the Lebesgue measure on R N . Hence we have So, it follows that For x = 0 and τ 0, using the chain rule and hypothesis H2 (iii), we have From (10) with τ = 0, we have From (11) and (9) it follows that Ω f η (z, u n )u n u n p dz −→ +∞.
Since u n ∈ N η for all n ∈ N, we have for some c 2 > 0. Comparing (12) and (13) we have a contradiction. Next we assume that y = 0. Let ξ 1 and introduce v n = (pξ) 1 p y n ∀n ∈ N.
From (15) and (18) we have d dt ϕ η (tu n ) t=t n = 0, so ϕ η (t n u n ), t n u n = 0) (by the chain rule and (18)), so t n u n ∈ N η ∀n n 2 , with 0 < t n < 1. However, we also have u n ∈ N η . Then on account of (19) and Proposition 4, we have a contradiction.
On account of hypotheses H2 (i),(ii), we can find c 3 > 0 such that Then from Proposition 5, we infer the following corollary.
Next we show that the elements of the Nehari manifold maximize the fibering function.
Choosing ε ∈ (0, λ 1 (q)c 9 ), we have for some c 10 > 0. Since p < r, it follows that Then from (21) and (23), it follows that k η u has a local maximizer t u > 0 and so we have Since u ∈ N η , from Proposition 4 we infer that We show that the elements of the Nehari manifold not only are nontrivial but in fact are bounded away from zero in norm.

Proposition 7.
If hypotheses H1, H2 hold, then there exists d 0 > 0 such that d 0 u r , u for all u ∈ N η .

Multiple Solutions for Problem
We show that both infima are realized.

Proposition 9.
If hypotheses H1, H2 hold, then there exist u * η ∈ N η + and u η ∈ N η − such that Proof. We consider a minimizing sequence {u n } n∈N ⊆ N η Corollary 1 implies that the sequence {u n } n∈N ⊆ W 1,ϑ 0 (Ω) is bounded. So, by passing to a subsequence if necessary, we may assume that u n w −→ u * η in W 1,ϑ 0 (Ω) and u n −→ u * η in L r (Ω).
Since u n ∈ N η + ⊆ N η , we have a (Du n ) + Du n q q = Ω f η (z, u n )u n dz, u n 0 ∀n ∈ N.
Then, using (38), (39) and Proposition 6, we have Arguing as in the proof of Proposition 10, we show that N η 0 is a natural constraint for the functional ϕ η , that is, y * η ∈ K ϕ η . In this case we replace the interval D + by the square and the functionals ζ, λ 0 , λ 1 by Reasoning as in the proof of Proposition 10, via the quantitative deformation lemma of Willem ([14], p. 38), we obtain the following result showing that N η 0 is a natural constraint for the functional ϕ η . Proposition 13. If hypotheses H1 and H2 hold and η > 0, then and so y * η ∈ W 1,ϑ 0 (Ω) ∩ L ∞ (Ω) is a nodal solution of (P η ).
Summarizing the situation for problem (P η ), we can state the following multiplicity result.

Proposition 14.
If hypotheses H1 and H2 hold and η > 0, then problem (P η ) has at least three nontrivial solutions
Proof. Let η n → 0 + . According to Proposition 14, we can find u n = u η n ∈ N η n 0 such that (see Proposition 15).
Claim. The sequence {u n } n∈N ⊆ W 1,ϑ 0 (Ω) is bounded. We argue by contradiction. So, suppose that at least for a subsequence, we have Let v n = u n u n for n ∈ N. Then v n = 1 for all n ∈ N and so we may assume that First suppose that v = 0. Using (42) and Proposition 6, for k > 1, we have (since k > 0, q < p, v n = 1; see Proposition 1). We pass to the limit as n → +∞ and use (44) (recalling that v = 0). Then m * k q p > 0 (see (42)).
Since k > 1 is arbitrary, we let k → +∞ and have a contradiction. Next suppose that v = 0 and set We have | Ω| N > 0 (here | · | N denotes the Lebesgue measure on R N ) and |u n (z)| → +∞ for a.a. z ∈ Ω. From (42) we have 0 < m η n 0 u n p = ϕ η n (u n ) u n p 1 p a (Dv n ) + for all n n 1 (see (43) and recall that q < p). Hypothesis H2 (ii) implies that Ω F(z, u n ) u n p dz −→ +∞ and so Ω F(z, u n ) u n p dz −→ +∞.
Passing to the limit as n → +∞ in (45), we have a contradiction. This proves the Claim.
On account of the Claim, we may assume that u n w −→ y * in W 1,ϑ 0 (Ω) and u n −→ y * in L r (Ω).

Conclusions
We have extended the classical multiplicity result to superlinear problems (see [10,25], three solutions theorem) to double-phase problems, under more general conditions on the reaction and we provided sign information for all the solutions produced.