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

Yang, Jie; Chen, Haibo; Liu, Senli

doi:https://doi.org/10.1155/2020/8237492

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Research Article | Open Access

Volume 2020 | Article ID 8237492 | https://doi.org/10.1155/2020/8237492

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

Jie Yang,^1,2Haibo Chen ,¹and Senli Liu¹

Academic Editor: Pietro d'Avenia

Received25 Nov 2019

Accepted03 Feb 2020

Published23 Mar 2020

Abstract

We consider the following double phase problem with variable exponents: . 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.

1. Introduction and Statement of Results

In this paper, we deal with the existence and multiplicity of solutions for the following double phase problem where is a real parameter, is a bounded domain with smooth boundary, , , and are Lipschitz continuous in . Moreover, and we also assume that the nonlinearity satisfies the following conditions:

is a Carathéodory function and there exists such that for all , where ., uniformly for a.e. . , uniformly for a.e. , where . There exists a constant such that for any or , where . There exists such that is nondecreasing in when and nonincreasing in for all ., for all and .

Remark 1. We point out that the condition is weaker than . It is not difficult to check that the condition is equivalent to the following condition (see [1]): is increasing in and decreasing in for all .
Hence, implies .

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 switches between two different elliptic behaviors have been intensively studied in recent years, see [2–11]. Recently, Mingione et al. have obtained the regularity theory for minimizers of (5), see, e.g., [7].

When and , problem becomes a -Laplacian problem of the form where . 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 , then . Vetro [12] studied the following Dirichlet boundary value problem involving the -Laplacian-like operator: where is the -Laplacian-like. They have established the existence and multiplicity results for the problem (7) when is sufficiently small.

In the particular case of ,, such problems have been recently studied in, e.g., [13–16]. The existence and multiplicity of weak solutions of problem with 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 . In [14], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions. Usually, the authors in those references considered the nonlinearities satisfying the Ambrosetti-Rabinowitz type condition ((AR) in short): i.e., there exist , , such that for and a.e. ,

Under some appropriate assumptions, one can consider a much weaker condition on

This means that is -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, 17–19] and the references therein. However, in this paper, we consider the problem in the case when the nonlinearity is -superlinear at both infinity and origin (see conditions and ). These conditions are weaker than the (AR) condition. For example, Papageorgiou, Vetro, and Vetro [16] investigated the following (,2)-equation with combined nonlinearities: where , be a bounded domain with a ²-boundary . Using the critical point theory, critical groups, and flow invariance arguments, the authors obtained at least five nontrivial smooth solutions of (11) when is ()-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 case. Moreover, can we assume that the nonlinearity satisfies a more natural and weaker -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 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 , we denote and we denote by the fact that

The letters ,,, 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 is that is a solution of if

It is formulated in a suitable Orlicz-Sobolev space that will be introduced in Section 2. It is easy to see that solutions of correspond to the critical points of the energy functional defined by where .

Now, we present the main results of this paper as follows:

Theorem 2. Suppose are satisfied. Then problem has at least one nontrivial weak solution in for all .

Theorem 3. Suppose are satisfied. Then there exists such that for all , problem has at least one solution and

Theorem 4. Suppose are satisfied. Then problem has infinitely many solutions in for all .

Theorem 5. Suppose and the following condition is a Carathéodory function, and there exist positive constants such that for all and , where such that with . Then problem has infinitely many solutions in for all .

Theorem 6. Suppose are satisfied. Then for all , problem has infinitely many solutions such that .

Theorem 7. Suppose are satisfied. Then for all , problem has infinitely many solutions such that ,.

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 case. Compared with [13], the main difficulty is that since both and are nonconstant functions, then 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 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.

2. Preliminaries

In order to discuss the problem , we need some theories on generalized Orlicz spaces and Sobolev spaces. For more details, we refer to the references [20–23]. The variable exponent Lebesgue space is defined by endowed with the Luxemburg norm

Note that, if is a constant function, the Luxemburg norm coincide with the standard norm of the Lebesgue space . Then, (,) becomes a Banach space, and we call it the variable exponent Lebesgue space. It is easy to check that the embedding is continuous, where and , are variable exponents such that in .

The following property of spaces with variable exponent is essentially due to Fan and Zhao [24].

Lemma 10. The space is a separable, uniformly convex Banach space, and its dual space is where . For any and , we have

The Musielak-Orlicz space is defined by endowed with the norm where is defined in (5). The space is a separable, uniformly convex, and reflexive Banach space. We denote by the space of all measurable functions with the seminorm

It is easy to check that the embeddings are continuous. Since whenever , we have

The related Sobolev space is defined by equipped with the norm where . The completion of in is denoted by and it can be equivalently renormed by via a Poincaré-type inequality, cf ([6], Proposition 2.18(iv)), under assumption (2). The spaces and are uniformly convex, and hence reflexive, Banach space. By (27), We have

We point out that if and for all , then is continuous. This embedding is compact if

Let us now define as and we denote the derivative operator by , that is , with

Here, denotes the dual space of , and denotes the paring between and . In the following lemma, we summarize some properties of , useful to study our problem. When , we refer to ([13], Proposition 3.1).

Lemma 11 (see [19], Lemma 3.4). Under the condition (2), is a mapping of type , that is, if in and , then in .

Lemma 12 (see [19], Lemma 3.2). Under the condition , is well defined on , and with Fréchet derivate given by Firstly, we show the functional satisfies the condition.

Lemma 13. If hypotheses ,, and hold, then satisfies the condition.

Proof. For every , let be a sequence, that is,

We claim that {} is bounded in . In fact, suppose by contradiction that , as Let . Up to a subsequence, we may assume that

We know that satisfies the following alternative: or . In what follows, we will show that under the condition , satisfies neither nor . This is a contradiction. Thus, is bounded.

If , then a.e. , as . Since is continuous in , for each , there exists such that

It is easily seen that and . If , then , which implies

Moreover, if , then, from(36) we have . So, we always have

Let be a sequence of positive real numbers such that for any and . Then for any and . Fix , using , (37), and the Lebesgue dominated convergence theorem we deduce that

Recall that as . So, we have or for large enough. Hence, from (31) and (38), we deduce that for any large enough. By combing this inequality with (41), as , we have

On the other hand, using condition and (40), for all large enough, we obtain

From (43) and (44), we obtain a contradiction. This shows that , and thus,

Let . It implies that

Using condition , we obtain

Also by and , we can get a constant such that

Thus, we get

From (31), we see that which implies

Similarly, from (31), we also get which implies for large enough.

We claim that Indeed, suppose by contradiction , then by (47)–(53) and Fatou’s lemma, we obtain which yields a contradiction. Therefore the sequence is bounded in . Thus, there is a subsequence (which we still denote by ) that converges weakly to some and strongly in . It is easy to check from () and Hölder’s inequality that

Then

So follows from Lemma 11.

3. Proofs of Theorems 2 and 3

First, we will show the functional satisfies the mountain pass geometry [25].

Lemma 14. Assume hypotheses hold. Then the functional satisfies the following properties: (i)There exist such that for any with (ii)There exists a such that .

Proof. Let us check (i). For any and small, it follows from that there exists such that Thus, for and , we have by the Sobolev embedding and . Since and arbitrarily small, there exist and such that for . Hence item (i) holds.
Let us check (ii). From , for any , we can choose a constant such that Then, for and , we deduce that If is large enough such that conclusion (ii) follows.

Proof of Theorem 2. Since the functional has the mountain pass geometry and satisfies the condition, the mountain pass theorem [25] gives that there exists a critical point . Moreover, , so is a nontrivial solution.

Lemma 15. Assume holds. Then there exist positive constants and such that and when .

Proof. Let with It follows from that there exists such that for all . Hence, we obtain Let where . Hence, we get for small enough. Therefore, substituting in (63), we see that Let us define From , we get that there exist small enough such that for all and as .

Proof of Theorem 3. By Lemma 13, satisfies the condition. Now in view of Lemma 13 and Lemma 15 and Lemma 14(ii) we can apply the mountain pass theorem to obtain a nontrivial critical point for such that On the other hand, from (62), we have Taking the limit in (66) and using Lemma 15, one has

4. Proofs of Theorems 4 and 5

Lemma 16. Assume the hypotheses hold. Then the functional satisfies the following properties: (i)There exist constants , such that for any with (ii)For each finite dimensional subspace , there exists an such that

Proof. As in the proof of Lemma 14, it is immediate to see that the case (i) is true. Let and be fixed. From (59), we obtain for all norms on are equivalent. Then, we can choose large enough such that . Therefore, we see that as , and the step is proved by taking with large enough.

Proof of Theorem 4. According to our assumption (), is an even functional. By the Lemma 13, satisfies the condition. Together with the Lemma 16, we can apply a version of the mountain pass theorem (see [25], Theorem 9.12) to obtain an unbounded sequence of weak solutions of problem ().
We finalize the section presenting a relation between the genus of and the number of solutions of the problem , where is a -dimensional linear subspace of . We invoke Clark’s Theorem in [25], Theorem 9.1. The next result is a compactness result on problem which we will use later.

Lemma 17. Assume that condition holds, then (i) is bounded from below(ii) satisfies the (PS) condition.

Proof. (i) Using , and for , we obtain Hence, is coercive following immediately from the above expression and . Therefore, is bounded from below.
(ii) Suppose is a sequence for . Thus and in as It follows from (i) that is bounded in . Up to a subsequence, we may assume that Since and in , (see [26], Proposition 3.5), we get that It is easy to check from () and Hölder’s inequality that where . Then So follows from Lemma 11.

Proof of Theorem 5. Consider is a -dimensional linear subspace of . We claim if is sufficiently small. Indeed, by the equivalence of norms on , there exists a constant such that for with Therefore, by , for with If is small enough, we have that The last inequality shows for all . It is clear that is isomorphic to and is homeomorphic to in . Hence, we obtain In the proof of Lemma 17, it was already established that is bounded from below, satisfies the (PS) condition, and . Clearly, implies is even. Consequently, by Clark’s Theorem in [25] (Theorem 9.1), possesses at least distinct pairs of nontrivial solutions. Since is arbitrary, we obtain infinitely many nontrivial solutions.

5. Proofs of Theorems 6 and 7

In this section, we will show that () has infinitely many pairs of solutions by using the Fountain Theorem and Dual Fountain Theorem. Firstly, we need to recall some preliminary results. Since is a reflexive and separable Banach space, there are and such that

Then, we define

We will apply the following Fountain Theorem ([25], Theorem 3.6).

Lemma 18. Assume that is a Banach space, and let be an even functional. If, for every , there exists such that Then has an unbounded sequence of critical values.

To prove Theorems 6 and 7, the following lemma is needed.

Lemma 19. Assume that, for any . Let then

Proof. Obviously, and so Let satisfy Then, there exists a subsequence of (which we still denote by ) such that , and which implies , and thus, . Since , then in . Hence, we get

Proof of Theorem 6. Let . According to , is an even functional. As the proof of Lemma 13, it follows from and () that satisfies the condition. For every , we shall prove that there exist such that We first show that () holds. For any , we choose . From Lemma 19 and , we see that as . As before, we also have from (62) that which implies that
Afterwards, we demonstrate that () holds. Let and . From (59), we obtain for all norms on are equivalent. Then, we can choose large enough such that . Therefore, we see that as . Hence, there exists large enough such that . Therefore, let , we obtain that

For the proof of Theorem 7, we need the following definitions and results.

Definition 20. Let be a separable and reflexive Banach space, . We say that satisfies the condition (with respect to ()), if any sequence for which , for any , and , as , contains a subsequence converging to a critical point of .

We are now ready to prove the Theorem 7.

Proof of Theorem 7. According to the Dual Fountain Theorem ([25], Theorem 3.18), it suffices to prove that for every , there exist such that Firstly, we show that () holds. Let and . Then similar to the proof of (), we see that for all norms on are equivalent. Then, we can choose large enough such that . Therefore, we see that as . Hence, there exists large enough such that . Therefore, let , we obtain that We show that holds. As we have done in the proof of Theorem 6, For any choosing . From Lemma 19 and , we see that as . As before, we also have from (88) that which implies that there exists for all choosing such that First from and , we observe that By , there exists such that for all . Now we define the function by By the definition of , we have and they are weakly-strongly continuous. Consider From the compact embedding and Lemma 19, we have Let and . Then, from (89) and (90), we obtain Passing the limit in the above inequality, as , we achieve that which, together with (88), implies that
Let be any sequence in such that Then similar to the proof of Lemma 13, we see that is bounded in . Thus, there is a subsequence (which we denote by ) that converges weakly to some and strongly in . It is easy to check from () and Hölder’s inequality that

Claim 21.

If Claim 21 holds true, then

So follows from Lemma 11. Hence, satisfies the condition. In order to prove Claim 21, we invoke to choose such that strongly in . Since and in , (see [26], Proposition 3.5), we get that

Hence, we obtain

Therefore, the Claim holds true and we conclude that as We next show that To see this, taking , we have

We pass limit in the right side of (100) as to obtain

Therefore, satisfies the condition for every . The proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their valuable comments and suggestions. H. B. Chen is supported by the National Natural Science Foundation of China (No. 11671403); J. Yang is supported by the Scientific Research Fund of Hunan Provincial Education Department (No. 17C1263 and No. 19B450) and the Natural Science Foundation of Hunan Province of China (No. 2019JJ50473).

References

O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,” Journal of Differential Equations, vol. 245, no. 12, pp. 3628–3638, 2008.
View at: Publisher Site | Google Scholar
V. V. Zhikov, “On Lavrentiev’s phenomenon,” Russian Journal of Mathematical Physics, vol. 3, pp. 249–269, 1995.
View at: Google Scholar
V. V. Zhikov, “On some variational problems,” Russian Journal of Mathematical Physics, vol. 5, pp. 105–116, 1997.
View at: Google Scholar
P. Baroni, M. Colombo, and G. Mingione, “Harnack inequalities for double phase functionals,” Nonlinear Analysis: Theory, Methods & Applications, vol. 121, pp. 206–222, 2015.
View at: Publisher Site | Google Scholar
P. Baroni, M. Colombo, and G. Mingione, “Regularity for general functionals with double phase,” Calculus of Variations and Partial Differential Equations, vol. 57, no. 2, p. 62, 2018.
View at: Publisher Site | Google Scholar
F. Colasuonno and M. Squassina, “Eigenvalues for double phase variational integrals,” Annali di Matematica Pura ed Applicata (1923 -), vol. 195, no. 6, pp. 1917–1959, 2016.
View at: Publisher Site | Google Scholar
M. Colombo and G. Mingione, “Regularity for double phase variational problems,” Archive for Rational Mechanics and Analysis, vol. 215, no. 2, pp. 443–496, 2015.
View at: Publisher Site | Google Scholar
P. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” CBMS Regional Conference Series in Mathematics, vol. 65, 1986.
View at: Publisher Site | Google Scholar
N. S. Papageorgiou and C. Vetro, “Superlinear (p(z), q(z))-equations,” Complex Variables and Elliptic Equations, vol. 64, no. 1, pp. 8–25, 2019.
View at: Publisher Site | Google Scholar
C. Vetro and F. Vetro, “On problems driven by the -Laplace operator,” Mediterranean Journal of Mathematics, vol. 17, no. 1, p. 24, 2020.
View at: Publisher Site | Google Scholar
N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, “Double-phase problems with reaction of arbitrary growth,” Zeitschrift für Angewandte Mathematik und Physik, vol. 69, no. 4, p. 108, 2018.
View at: Publisher Site | Google Scholar
C. Vetro, “Weak solutions to Dirichlet boundary value problem driven by p(x)-Laplacian-like operator,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2017, no. 98, pp. 1–10, 2017.
View at: Publisher Site | Google Scholar
W. Liu and G. Dai, “Existence and multiplicity results for double phase problem,” Journal of Differential Equations, vol. 265, no. 9, pp. 4311–4334, 2018.
View at: Publisher Site | Google Scholar
W. Liu and G. Dai, “Three ground state solutions for double phase problem,” Journal of Mathematical Physics, vol. 59, no. 12, article 121503, 2018.
View at: Publisher Site | Google Scholar
K. Perera and M. Squassina, “Existence results for double-phase problems via Morse theory,” Communications in Contemporary Mathematics, vol. 20, no. 2, article 1750023, 2018.
View at: Publisher Site | Google Scholar
N. S. Papageorgiou, C. Vetro, and F. Vetro, “Multiple solutions with sign information for a -equation with combined nonlinearities,” Nonlinear Analysis, vol. 192, article 111716, 2020.
View at: Publisher Site | Google Scholar
A. Azzollini, P. d’Avenia, and A. Pomponio, “Quasilinear elliptic equations in via variational methods and Orlicz-Sobolev embeddings,” Calculus of Variations and Partial Differential Equations, vol. 49, no. 1-2, pp. 197–213, 2014.
View at: Publisher Site | Google Scholar
Y. Deng and W. Huang, “Ground state solutions for generalized quasilinear Schrödinger equations without (AR) condition,” Journal of Mathematical Analysis and Applications, vol. 456, no. 2, pp. 927–945, 2017.
View at: Publisher Site | Google Scholar
I. H. Kim and Y. H. Kim, “Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents,” Manuscripta Mathematica, vol. 147, no. 1-2, pp. 169–191, 2015.
View at: Publisher Site | Google Scholar
P. Harjulehto, P. Hästö, and R. Klén, “Generalized Orlicz spaces and related PDE,” Nonlinear Analysis: Theory, Methods & Applications, vol. 143, pp. 155–173, 2016.
View at: Publisher Site | Google Scholar
V. D. Radulescu and D. D. Repovs, Partial differential equations with variable exponents: variational methods and qualitative analysis, CRC Press, Boca Raton, 2015.
View at: Publisher Site
W. Xie and H. Chen, “Existence and multiplicity of solutions for -Laplacian equations in ,” Mathematische Nachrichten, vol. 291, no. 16, pp. 2476–2488, 2018.
View at: Publisher Site | Google Scholar
L. Xu and H. Chen, “Ground state solutions for quasilinear Schrödinger equations via Pohožaev manifold in Orlicz space,” Journal of Differential Equations, vol. 265, no. 9, pp. 4417–4441, 2018.
View at: Publisher Site | Google Scholar
X. Fan and D. Zhao, “On the spaces and ,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001.
View at: Publisher Site | Google Scholar
M. Williem, “Minimax theorems,” in Progress in nonlinear differential equations and their applications, vol. 24, Birkhäuser, Boston, 1996.
View at: Google Scholar
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011.
View at: Publisher Site

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Copyright © 2020 Jie Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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