Abstract
We consider the following parametric double-phase problem: . We do not impose any global growth conditions to the nonlinearity , which refer solely to its behavior in a neighborhood of . And we will show that they suffice for the multiplicity of signed and nodal solutions of the double-phase problem above when is large enough.
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 parameter, is a bounded domain with smooth boundary, , and we also assume that the nonlinearity satisfies the following conditions:
(f1) is a function, for
(f2) There exists such that uniformly for
(f3) There exists such that uniformly for , where
(f4) There exists a constant such that for and all
(f5) For the in ,
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 [1, 2]). The energy functionals of the form where the integrand which switches between two different elliptic behaviors has been intensively studied since the late eighties (see [3–9]). Recently, Colombo and Mingione in [7] have obtained the regularity theory for minimizers of (6).
The double-phase problem has been studied extensively recently. The existence of a sign-changing ground state solution to problem (7) has been proven by Liu and Dai in [3], when were assumed to satisfy the p-superlinear growth condition and Ambrosetti-Rabinowitz condition. In [4], by using Morse theory, Perera and Squassina obtained a nontrivial weak solution of problem (7), when . In [5], by utilizing the Nehari method, Liu and Dai obtained three ground state solutions.
Motivated by the above works, we intend to establish the multiplicity of both signed and nodal solutions of problem (), when is large enough. Here, we note that the assumptions (f1)–(f5) that we make on the nonlinearity refer only to its behavior in a neighborhood of .
We present the main result of this paper as follows:
Theorem 1. Suppose are satisfied. Then, there exists , and if , problem () has at least one positive solution, one negative solution, and a sign-changing solution.
Theorem 2. Suppose are satisfied. Then, there exists , and if , for any given , problem () has pairs of solutions with . Moreover, are pairs of sign-changing solutions.
Remark 3. From and , it can be seen that . We may think of with , which clearly satisfy .
We remark that [3, 5] obtained only one sign-changing solution. However, in Theorem 2, since is arbitrary, we get infinitely many sign-changing solutions. To the best of our knowledge, little has been done in the literature on the existence of multiple nodal solutions for the parametric Dirichlet problem with the minimal conditions on the nonlinearity .
The proof will be done by variational techniques. Since we have no information on the behavior of the nonlinearity at the infinity, we adapt the argument introduced by Costa and Wang [10], which consists in making a suitable modification on , solving a modified problem, and then checking that, for a large enough , the solutions of the modified problem are indeed solutions of the original one.
The paper is organized as follows. In Section 2, we modify the original problem and prove the Palais-Smale condition for the modified functional. We present some tools which are useful to establish a multiplicity result. In Section 3, we prove Theorem 1. Theorem 2 is proven in Section 4.
2. The Modified Problem
To prove our main results, we need to present the variational setting of our problem. Firstly, we introduce some notations and some necessary definitions. The Musielak-Orlicz space associated with the function consists of all measurable functions with the -modular
The Musielak-Orlicz space is defined by endowed with the norm
The space is a separable, uniformly convex, and reflexive Banach space. We denote by the norm in and 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 Poincare-type inequality (cf [6], Proposition 2.18(iv)), under assumption (2). The spaces and are uniformly convex and hence are reflexive, Banach spaces. By Proposition 2.15 in [6], we know that the Sobolev embedding is compact since . By (14), We have
From now on, we denote by for convenience in writing.
We first mention that imply that there exist , such that for and all .
Now, define the even cutoff function as
Moreover, for all , satisfies that
Let
Also, we set
From hypotheses , it is easy to check that is a Carathéodory function and satisfies the following properties.
Lemma 4. If hypotheses , and hold, then the functions and satisfy the following properties: (i)There exist such that for (ii)There exists such that for and
Proof. The proof is similar to that of Lemma 1.1 in [10].
Now let us consider the modified problem of ():
The corresponding energy functional of () is
We easily get that the functional , and its critical points are the solutions of (). We note that solutions of () with are also solutions of (). We shall search solutions of () as critical points of .
Let us now define as and we denote the derivative operator by , that is, with
In the following lemma, we summarize some properties of , useful to study our problem.
Lemma 5. Under the condition (2), is a mapping of type (); that is, if in and , then in .
Proof. The proof is similar to that of Proposition 3.1(ii) in [3].
Firstly, we show the functional satisfies the (PS) condition.
Lemma 6. If hypotheses – and hold, then satisfies the (PS) condition.
Proof. For every , let be a –sequence, that is,
We claim that is bounded in . Indeed, if , we have done. If , by Lemma 4(ii), then we have that where . Hence, is bounded. Therefore, there is a subsequence (which we still denote by ) that converges weakly to some and strongly in . It is easy to check from Lemma 4(i) and Hölder’s inequality that
Then,
So follows from Lemma 5.
Next, we will show the functional satisfies the Mountain Pass Geometry [11].
Lemma 7. Assume that hypotheses – are satisfied. Then, the satisfies the following conditions: (i)For every there exist , such that with (ii)There exists such that and for all
Proof. (i) It follows from Lemma 4(i) that there exists such that
Then, for all , we have
For , we can choose , then
(ii) Let , such that
Firstly, we note that So there exists such that for all ,
Then, by (20), (37), and (38), we get
This completes the proof of Lemma 7.
Lemma 8. Assume that hypotheses – and are satisfied, and , then there exists such that .
Proof. Let . We observe from Lemma 4(ii) that
Since , there exists such that
From Theorem 8.4 in [12], we say that the solutions of () enjoy the estimates given in the next lemma.
Lemma 9. Let be the weak solution to () under the assumptions –. Then, the following estimate holds: where
Now we are ready to show the existence of a solution to () for a large .
Lemma 10. Assume that conditions – hold. Then, for any , there exists a nontrivial solution of (), such that where .
Proof. By Lemma 6 and Lemma 7, we conclude that satisfies the (PS) condition and the Mountain Pass Geometry. Consequently, by the Mountain Pass Theorem (Theorem 2.2 in [13]), there exists a such that where .
From (38), we easily obtain that the function is continuous and . It follows from , for which is small enough, we get
Hence, there exists such that where . Then, we obtain where
Let . Obviously, , and then we get from (44) that
Therefore, there exists such that . Together with Lemma 8, we obtain
It follows from Lemma 9, we get that where recalling that .
3. Proof of Theorem 1
In this section, we prove our main result. We will show () has a positive solution, a negative solution, and a nodal solution. And the solutions obtained satisfy the estimate . This fact implies that these solutions are indeed solutions of the original problem ().
Proof of Theorem 1. Consider the following problem: where
Define the corresponding functional where Obviously, and satisfy all the conditions of Theorem 1. Let be a nontrivial critical point of , which implies that is a weak solution of (53). It is known by Lemma 4.1 of [5] that a.e. in . From Theorem 3.3 in [5], we conclude that . Therefore, by Lemma 10, where so there exists such that for all Thus, is also a nontrivial solution of original problem () for all .
Similarly, we can define where We also get a negative solution of our original problem () for all .
We next show that there is a sign-changing solution for large enough. We can apply the method introduced by Li and Wang [14] to our case. Since is a real, reflexive, and separable Banach space, there are and such that
For , we denote
On , we define a closed convex cone
And is a Banach space densely embedded in . Assume that has interior points in . As in Section 3 of [14], we may define a partial order relation in : . And we define . In order to apply the method introduced by Li and Wang in [14] (Example 3.2 and Corollary 3.2), we take on , and for , . Hence, we obtain and link.
It follows from Lemma 4(ii) that there exist such that
Then, for each , we obtain that for all norms on are equivalent. Since , we may get which is large enough such that for all and for all .
Using (34) and the Sobolev embedding for , we can choose some such that for . Let us define where . Therefore, by [14], we see that is a critical value of and has a sign-changing critical point at this critical value. Therefore, by Lemma 10, where so there exists such that for all . Thus, is also a nontrivial sign-changing solution of original problem () for all .
4. Proof of Theorem 2
In this section, we present a multiplicity result for the modified problem. We shall use arguments in [13] to get solutions for first. Next, we use proofs in [14] to get the nodal property of the solutions.
Proof of Theorem 2. By (63), we can choose such that for all and for all Set . Let . Let denote the family of sets such that is closed in and symmetric with respect to 0. For , defines the genus of . Then, for we can define where Now we can apply Proposition 9.30 in [13] to functional , we get that are all critical values of . And possesses at least pairs of distinct nontrivial solutions. Then, by Lemma 10, there exists , such that ; these pairs solutions of are also solutions of the original problem .
Secondly, we will obtain the nodal property of the solutions by Theorem 2.3 in [14]. We need a procedure similar to the one we used earlier to rule this out. Let
Let . For , also denotes the genus of . By (63), we can also choose such that for all and for all By (64), we can also choose such that for all and for all By Theorem 2.3 in [14], has at least pairs of critical points in , with critical values where Moreover, . Finally, by Lemma 10, where so there exists such that for all Thus, , are also nontrivial sing-changing solutions of the original problem () for all .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
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), the Natural Science Foundation of Hunan Province of China (No. 2019JJ50473), and the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 19B450).