Existence and Multiplicity of Solutions for a Class of Anisotropic Double Phase Problems

<jats:p>We consider the following double phase problem with variable exponents: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mfenced open="{" close=""><mml:mrow><mml:mtable class="cases"><mml:mtr><mml:mtd columnalign="left"><mml:mo>−</mml:mo><mml:mi mathvariant="normal">div</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:msup><mml:mrow><mml:mfenced open="|" close="|"><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>a</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mfenced open="|" close="|"><mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mtext> </mml:mtext><mml:mtext>in</mml:mtext><mml:mtext> </mml:mtext><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="left"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mtext>on</mml:mtext><mml:mtext> </mml:mtext><mml:mi>∂</mml:mi><mml:mi>Ω</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math>. By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. Moreover, infinitely many pairs of solutions are provided by applying the Fountain Theorem, Dual Fountain Theorem, and Krasnoselskii’s genus theory.</jats:p>

Similar problems have been investigated and it is well known they have a strong physical meaning because they appear in the models of strongly anisotropic materials, see, e.g., [2,3]. The energy functionals of the form where the integrand H switches between two different elliptic behaviors have been intensively studied in recent years, see [2][3][4][5][6][7][8][9][10][11]. Recently, Mingione et al. have obtained the regularity theory for minimizers of (5), see, e.g., [7]. When aðxÞ = 1 and λ = 1, problem ðP λ Þ becomes a ðpðxÞ, qðxÞÞ-Laplacian problem of the form where −Δ pðxÞ u≔− div ðj∇uj pðxÞ−2 ∇uÞ. In particular, we refer to [9] where the authors proved the existence of one and three nontrivial weak solutions of (6), by the mountain pass theory and Morse theory. If pðxÞ = qðxÞ, then aðxÞ = 1. Vetro [12] studied the following Dirichlet boundary value problem involving the pðxÞ-Laplacian-like operator: is the pðxÞ-Laplacian-like. They have established the existence and multiplicity results for the problem (7) when λ is sufficiently small. In the particular case of pðxÞ ≡ p,qðxÞ ≡ q, such problems have been recently studied in, e.g., [13][14][15][16]. The existence and multiplicity of weak solutions of problem ðP λ Þ with λ = 1 has been established in Liu and Dai [13]. In [15], by using the Morse theory, Perera and Squassina obtained a nontrivial weak solution of problem ðP λ Þ. In [14], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions. Usually, the authors in those references considered the nonlinearities f ðx, tÞ satisfying the Ambrosetti-Rabinowitz type condition ((AR) in short): i.e., there exist L > 0, θ > q, such that for |t| ≥ L and a.e.
x ∈ Ω, Under some appropriate assumptions, one can consider a much weaker condition on f ðx, tÞ This means that F is q-superlinear at infinity. But the (AR) condition is useful and natural to ensure the mountain pass geometry and the Palais-Smale condition ((PS) in short). So it have attracted much interest in recent literature, see for example [13,15,[17][18][19] and the references therein. However, in this paper, we consider the problem ðP λ Þ in the case when the nonlinearity F is q + -superlinear at both infinity and origin (see conditions ð f 2 Þ and ð f 3 Þ). These conditions are weaker than the (AR) condition. For example, Papageorgiou, Vetro, and Vetro [16] investigated the following (p,2)-equation with combined nonlinearities: where λ > 0, 2 < p<+∞,Ω ⊂ ℝ N , be a bounded domain with a C 2 -boundary ∂Ω. Using the critical point theory, critical groups, and flow invariance arguments, the authors obtained at least five nontrivial smooth solutions of (11) when f is (p − 1)-superlinear near ±∞ but does not satisfy the (AR) condition. Now, a natural question is whether the results contained in [13] can be generalized to the variable exponents ðpðxÞ, q ðxÞÞ case. Moreover, can we assume that the nonlinearity f satisfies a more natural and weaker ðq + − 1Þ-superlinear condition near ±∞ instead of the (AR) condition? Inspired by the above works, we will answer these questions. For a detailed motivation of our context and additional references, we refer to the introduction of [8,20]. To the best of our knowledge, there are very few papers related to the existence of solutions of problem ðP λ Þ with variable exponents. This paper was motivated by the interest in applications of the variable exponent Orlicz-Sobolev spaces. Before stating our main results, we introduce some notations.
1.1. Notations and definitions. Throughout this paper, we define the class For any p ∈ C + ð ΩÞ, we denote 2 Advances in Mathematical Physics and we denote by p 1 ≪ p 2 the fact that The letters C,C i ,i = 1, 2, ⋯, denote positive constants which may vary from line to line but are independent of the terms which will take part in any limit process. The notion of weak solution for problem ðP λ Þ is that It is formulated in a suitable Orlicz-Sobolev space W 1,ℋ 0 ðΩÞ that will be introduced in Section 2. It is easy to see that solutions of ðP λ Þ correspond to the critical points of the energy functional I λ defined by where Fðx, tÞ = Ð t 0 f ðx, sÞdx. Now, we present the main results of this paper as follows: for all x ∈ Ωand t ≥ 0, where β ∈ Cð ΩÞ such that 1 < βðxÞ < p * ðxÞ with β + < p − . Then problem ðP λ Þ has infinitely many solutions in W 1,ℋ 0 ðΩÞ for all λ > 0.
Remark 8. Note that our Theorems 2-7 answer the above questions. To be precise, Theorems 2, 4, 6, and 7 extend the main results of [13] to the variable exponents ðpðxÞ, qðxÞÞ case. Compared with [13], the main difficulty is that since both pðxÞ and qðxÞ are nonconstant functions, then ðP λ Þ has a more complicated structure, due to its nonhomogeneities and to the presence of the nonlinear term.
Remark 9. In Theorem 5, we obtain infinitely many solutions by using Krasnoselskii's genus theory. Moreover, we consider continuous functions f = f ðx, uÞ satisfying the growth condition The rest of this paper is organized as follows. In Section 2, we state some preliminary notations and the main lemmas. In Section 3, we prove the Theorems 2 and 3. The proofs of Theorems 4-5 are given in Section 4. By using the Fountain Theorem and the Dual Fountain Theorem, infinitely many pairs of solutions are provided in Section 5.

Preliminaries
In order to discuss the problem ðP λ Þ, we need some theories on generalized Orlicz spaces and Sobolev spaces. For more details, we refer to the references [20][21][22][23]. The variable exponent Lebesgue space L pðxÞ ðΩÞ is defined by endowed with the Luxemburg norm Note that, if p is a constant function, the Luxemburg norm kuk pð⋅Þ coincide with the standard norm kuk p of the Lebesgue space L p ðΩÞ. Then, (L pðxÞ ðΩÞ,kuk pð⋅Þ ) becomes a Banach space, and we call it the variable exponent Lebesgue space. It is easy to check that the embedding L p 2 ðxÞ ðΩÞ↪ L p 1 ðxÞ ðΩÞ is continuous, where 0 < |Ω|<∞ and p 1 ,p 2 are variable exponents such that p 1 ≤ p 2 in Ω.
The following property of spaces with variable exponent is essentially due to Fan and Zhao [24].
Lemma 10. The space ðL pð⋅Þ ðΩÞ, k⋅k pð⋅Þ Þ is a separable, uniformly convex Banach space, and its dual space is L p′ð⋅Þ ðΩÞ where ð1/pðxÞÞ + ð1/p ′ ðxÞÞ = 1. For any u ∈ L pð⋅Þ ðΩÞ and v ∈ L p ′ ð⋅Þ ðΩÞ, we have 3 Advances in Mathematical Physics The Musielak-Orlicz space L H ðΩÞ is defined by endowed with the norm where ℋ is defined in (5). The space L H ðΩÞ is a separable, uniformly convex, and reflexive Banach space. We denote by L qð⋅Þ a ðΩÞ the space of all measurable functions u : It is easy to check that the embeddings are continuous. Since ρℋ ðu/kukℋ Þ = 1 whenever u ≠ 0, we have The related Sobolev space W 1,H ðΩÞ is defined by equipped with the norm where k∇uk ℋ = jk∇ukj ℋ . The completion of C ∞ 0 ðΩÞ in W 1,H ðΩÞ is denoted by W 1,H 0 ðΩÞ and it can be equivalently renormed by We point out that if r ∈ C + ð ΩÞ and rðxÞ ≤ p * ðxÞ for all Let us now define Jð⋅Þ: and we denote the derivative operator by A, that is Firstly, we show the functional I λ satisfies the ðCÞ c condition.

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Proof. For every c ∈ ℝ, let fu n g ⊂ W 1,H 0 ðΩÞ be a ðCÞ c − sequence, that is, We claim that {u n } is bounded in W 1,ℋ 0 ðΩÞ. In fact, suppose by contradiction that ∥u n ∥⟶ + ∞, as n ⟶ ∞: Let v n = u n /∥u n ∥, n ≥ 1. Up to a subsequence, we may assume that v n ⟶ v, a:e: in Ω, We know that v satisfies the following alternative: v = 0 or v ≠ 0. In what follows, we will show that under the condition ∥u n ∥⟶ + ∞, v satisfies neither v = 0 nor v ≠ 0. This is a contradiction. Thus, fu n g is bounded.
If v = 0, then v n ⟶ 0 a.e. x ∈ Ω, as n ⟶ ∞. Since I λ ðtu n Þ is continuous in t ∈ ½0, 1, for each n, there exists t n ∈ ½0, 1ðn = 1, 2,⋯Þ such that It is easily seen that t n > 0 and I λ ðt n u n Þ ≥ c > 0 = I λ ð0Þ = I λ ð0u n Þ. If t n < 1, then ðd/dtÞI λ ðtu n Þj t=t n = 0, which implies Moreover, if t n = 1, then, from(36) we have hI λ ′ðu n Þ, u n i = o n ð1Þ. So, we always have Let γ k be a sequence of positive real numbers such that γ k > 1 for any k and lim k→+∞ γ k = +∞. Then ∥γ k v n ∥ = γ k > 1 for any k and n. Fix k, using ð f 1 Þ, (37), and the Lebesgue dominated convergence theorem we deduce that Recall that ∥u n ∥⟶ + ∞ as n ⟶ ∞. So, we have ∥u n ∥>γ k or 0 < γ k /∥u n ∥<1 for n large enough. Hence, from (31) and (38), we deduce that for any n large enough. By combing this inequality with (41), as n, k⟶+∞, we have On the other hand, using condition ð f 4 Þ and (40), for all n large enough, we obtain From (43) and (44), we obtain a contradiction. This shows that v ≠ 0, and thus, v n x ð Þ ⟶ v x ð Þ ≠ 0 a:e: in Ω: Let Ω ≠ ≔ fx ∈ Ω : vðxÞ ≠ 0g. It implies that Using condition ð f 3 Þ, we obtain Also by ð f 1 Þ and ð f 3 Þ, we can get a constant C 2 > 0 such that

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Thus, we get From (31), we see that which implies Similarly, from (31), we also get which implies for n large enough. We claim that |Ω ≠ | = 0: Indeed, suppose by contradiction |Ω ≠ | ≠ 0, then by (47)-(53) and Fatou's lemma, we obtain which yields a contradiction. Therefore the sequence fu n g is bounded in W 1,H 0 ðΩÞ. Thus, there is a subsequence (which we still denote by fu n g) that converges weakly to some u ∈ W 1,H 0 ðΩÞ and strongly in L αð·Þ ðΩÞ. It is easy to check from (f 1 ) and Hölder's inequality that Then So u n ⟶ u follows from Lemma 11.

Proofs of Theorems 2 and 3
First, we will show the functional I λ satisfies the mountain pass geometry [25].
Proof of Theorem 2. Since the functional I λ has the mountain pass geometry and satisfies the ðCÞ c condition, the mountain pass theorem [25] gives that there exists a critical point u ∈ W 1,H 0 ðΩÞ. Moreover, IðuÞ = c ≥ α > 0 = Ið0Þ, so u is a nontrivial solution.

Proofs of Theorems 4 and 5
Lemma 16. Assume the hypotheses ðf 1 Þ − ð f 3 Þ hold. Then the functional I λ satisfies the following properties: (i) There exist constants ρ, δ > 0, such that I λ ðuÞ ≥ δ for any u ∈ W 1,H 0 ðΩÞ with ∥u∥ = ρ (ii) For each finite dimensional subspaceX ⊂ W 1,H 0 ðΩÞ, there exists an R = RðXÞ such that I λ ≤ 0, onX \ B R ðXÞ: Proof. As in the proof of Lemma 14, it is immediate to see that the case (i) is true. Let e ∈X and ∥e∥ = 1 be fixed. From (59), we obtain

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for all norms onX are equivalent. Then, we can choose M large enough such that 1/p − − λMC 9 < 0. Therefore, we see that I λ ðteÞ ⟶ −∞, as n ⟶ ∞, and the step is proved by taking v 0 = t 0 e with t 0 > R large enough.
Proof of Theorem 4. According to our assumption (f 5 ), I λ is an even functional. By the Lemma 13, I λ satisfies the ðCÞ c condition. Together with the Lemma 16, we can apply a Z 2 version of the mountain pass theorem (see [25], Theorem 9.12) to obtain an unbounded sequence of weak solutions of problem (P λ ). We finalize the section presenting a relation between the genus of K and the number of solutions of the problem ðP λ Þ, where K is a k-dimensional linear subspace K ⊂ C ∞ 0 ðΩÞ of W 1,H 0 ðΩÞ. We invoke Clark's Theorem in [25], Theorem 9.1. The next result is a compactness result on problem ðP λ Þ which we will use later.

Lemma 17.
Assume that condition ð f 6 Þ holds, then (i) I λ is bounded from below (ii) I λ satisfies the (PS) condition.
Proof of Theorem 5. Consider K is a k-dimensional linear subspace K ⊂ C ∞ 0 ðΩÞ of W 1,H 0 ðΩÞ. We claim I λ j K < 0 if ∥u∥ ≤r < 1 is sufficiently small. Indeed, by the equivalence of norms on K, there exists a constant C 12 > 0 such that C 12 ∥u ∥ β + ≤ Ð Ω juj βðxÞ dx for u ∈ K with ∥u∥≤1: Therefore, by ð f 6 Þ, for u ∈ K with ∥u∥<1: If r ∈ ð0, 1Þ is small enough, we have that The last inequality shows I λ j K < 0 for all u ∈ S k r = fu ∈ K : ∥u∥ = rg. It is clear that K is isomorphic to ℝ k and S k r is homeomorphic to S k−1 in ℝ k . Hence, we obtain γðS k r Þ = k: In the proof of Lemma 17, it was already established that I λ ∈ C 1 ðX, ℝÞ is bounded from below, satisfies the (PS) condition, and I λ ð0Þ = 0. Clearly, ð f 5 Þ implies I λ is even. Consequently, by Clark's Theorem in [25] (Theorem 9.1), I λ possesses at least k distinct pairs of nontrivial solutions. Since k is arbitrary, we obtain infinitely many nontrivial solutions.

Proofs of Theorems 6 and 7
In this section, we will show that (P λ ) has infinitely many pairs of solutions by using the Fountain Theorem and Dual Fountain Theorem. Firstly, we need to recall some Then, we define We will apply the following Fountain Theorem ( [25], Theorem 3.6).

Lemma 18.
Assume that X is a Banach space, and let φ ∈ C 1 ðX, RÞ be an even functional. If, for every k ∈ ℕ, there exists ρ k > r k > 0 such that Then φ has an unbounded sequence of critical values.
To prove Theorems 6 and 7, the following lemma is needed.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.