Abstract
In this paper, we consider a class of fractional -Laplacian problems with critical and negative exponents. By decomposition of the Nehari manifold, the existence and multiplicity of nontrivial solutions for the above problems are established with respect to a sufficiently small parameter.
1. Introduction and Main Result
Laplace transformation is an integral transformation commonly used in engineering mathematics. The transformation is a linear transformation that transforms a function with a real parameter number into a function with complex parameters. The Laplace transform has extensive applications in many fields of engineering technology and scientific research, especially when it plays an important role in mechanical systems, electrical systems, automatic control systems, reliability systems, and random service systems. In circuit analysis, it is often necessary to solve the differential equation or the integral equation, which can be solved by the Laplace transformation. The application of nonlinear equations promoted the development of nonlinear sensitive electronic devices on the load side and grid side of the power system. The stable operation of the power system at each level can be effectually protected by exploring the nonlinear phenomena in the case of ferromagnetic resonance overvoltage situation. So, studying the Laplacian system is an important topic.
In this paper, we study the following fractional -Laplacian system: where is a bounded domain in , with , , with , where is the fractional critical Sobolev exponent, and is a parameter. and satisfy the conditions such that , where . The fractional -Laplacian operator is defined as
We define where with and . The space is equipped with the norm , where
Set with the norm . is called a weak solution of problem (1) if for all . The best fractional critical Sobolev constant is defined as
In recent years, fractional Laplacian and -Laplacian systems with subcritical and critical nonlinearities have been studied widely. Chen and Deng [1] and Li and Yang [2] studied the following critical fractional Laplacian system with a lower-order term: where is a bounded domain with a smooth boundary, , , satisfy , where and , are parameters. The main difficulty lies in finding the interval of where the local condition is satisfied. The authors both adopted the explicit formula of extremal function related to the best Sobolev constant and some useful estimates established in Barrios et al. [3] (Lemma 3.8) to overcome this difficulty. Compared with the fractional Laplacian system with critical nonlinearities, for the critical fractional -Laplacian system with , we must face the difficulty that the explicit formula for minimizers of critical Sobolev constant does not exist. Chen and Squassina [4] overcame this difficulty by borrowing the asymptotic estimates for minimizers of , which were obtained in Brasco et al. [5].
On the other hand, much attention has been focused on discussing the fractional -Laplacian system with negative exponent and subcritical nonlinearity. In [6], Goyal first investigated the following fractional Laplacian system with sign-changing nonlinearity: where , , , and is a sign-changing function. Using the decomposition of the Nehari manifold, the multiplicity of positive solutions for (8) with respect to the pair of parameters was established. Furthermore, the author extended the above same result to the following -fractional Laplacian system: where , , , is a sign-changing function. Very recently, Saoudi [7] investigated the following fractional -Laplacian system: where , , . Using the variational method, the author proved that (10) has at least two positive solutions when the pair of parameters satisfies certain conditions. We have found that it is easier to deal with subcritical problems than critical ones because compact conditions for Sobolev embedding are satisfied in the subcritical case, while global condition for the energy functional corresponding to (1) does not usually hold in the case of critical problems. Moreover, Arora and Fiscella [8] studied a class of double-phase problems with a negative exponent and a critical Sobolev nonlinearity. The proof of the main result is based on a suitable minimization argument on the Nehari manifold. Zuo et al. [9] combined the effects of a nonlocal operator with critical nonlinearity. Suitable embedding results were developed to establish the existence of infinitely many solutions and to provide an estimate of the boundedness of these solutions. For more recent results on fractional -Laplacian problems with singular terms and critical terms simultaneously, see [10–13] and references therein. In fact, Ghanmi et al. [10] studied a class of nonlocal -Kirchhoff problems with a negative exponent and critical nonlinearity. The authors established the multiplicity of positive solutions to the above problems by using a truncation argument. Sang [11] considered a fractional critical system with -Laplacian operator and negative exponents. By applying fibering map analysis, the existence of two positive solutions for the above systems was obtained. Furthermore, Saoudi et al. [12] proved the existence of solutions to a nonlocal problem with a singular term and a discontinuous critical nonlinearity. Fiscella et al. [13] investigated the existence of nontrivial solutions for critical systems driven by the fractional -Laplacian operator. The main features of this paper are the presence of critical nonlinearities and singular terms.
Motivated by the above results, we consider fractional critical -Laplacian system (1). We combine critical problems with negative exponents. Note that the energy functional corresponding to (1) is not differentiable in the sense of Gâteaux; the method used in [1, 2, 4, 14–20] cannot be applied to our problem directly. Since , we cannot also extend the methods used in a single equation with critical and negative exponents [21, 22] when to problem (1). We use the concentration compactness principle [23, 24] to avoid this barrier. Our idea comes from Wang et al. [25].
The energy functional associated with problem (1) is defined by
We define a set and decompose with the following subsets:
Our main result is the following.
Theorem 1. There exists such that for every , problem (1) has at least two nontrivial solutions and in . More specifically, is a local minimizer of in with , and is a minimizer of on with .
2. Preliminaries and Some Lemmas
Let
and the relationship between and has been revealed in [4]. In order to prove our main result, the following lemmas are needed.
Lemma 2. The functional has a local minimum in with .
Proof. Since for every , where we have used the Hölder inequality, (6) and (14). We come to Hence, there exist and such that is bounded on for every . It follows that is well defined for fixed . Furthermore, choosing with all , , we have thus, for all , and small enough. Consequently, .
Lemma 3. There exists such that .
Proof. The definition of tells us that there exists a minimizing sequence such that . We assume that and . By , there is a subsequence, which is still denoted by , such that
In terms of the fractional concentration compactness principle [10] (Theorem 2.5), there exist two Borel regular measures and , denumerable, , , with , such that
It follows from Vitali’s theorem that
which, coupled with (20) and (21), gives
If , then
Combining with the definition of , we deduce that
In the following, we focus on showing . Firstly, by (21), we have , so we only need to prove that . We assume by contradiction that . Since , we derive that
If , for all , it follows that
which is wrong if we choose .
If there exists a subsequence such that , then
where with , which is a contradiction. Set , we have that . This completes the proof of Lemma 3.
Lemma 4. is coercive in .
Proof. For , we have it follows from that is coercive on .
Let
Lemma 5. There exist two and only two numbers and with such that for all and .
Proof. For , we define by
Let , which yields
Therefore, achieves its maximum at , and
Note that if and only if . By the condition that , we deduce that and . A simple computation leads to
Consequently, has exactly two zero points and with such that .
If and , then
so .
Furthermore, implies that
by the definition of , we have
namely, . Therefore, . Similarly, we get .
Lemma 6. For , we have .
Proof. Assume that . If , then By (39) and (40), we get Combining with (41) and (42), we derive that which contradicts with .
Lemma 7. Assume that , then is a closed set in -topology.
Proof. Assume that and strongly in . In the following, we prove that . For , we have and so, Therefore, . We can show that by contradiction. Otherwise, if , we get that from the definition of that By (34), (46), and (47), we deduce that which contradicts with for all .
Lemma 8. Let , then for every , there exist a number and a continuous function such that
Proof. We define as follows: For , we have We deduce that there exists such that has a unique continuous solution for , by using the implicit function theorem at the point . Since , then . By for , , we obtain that is, Since and we can choose sufficiently small such that for every with , that is, This completes the proof of Lemma 8.
Lemma 9. There exists a constant such that for each and all .
Proof. Assume that there exists such that ; thus, Equations (14) and (15) lead On the other hand, from (6) and (14), we have Choosing we have which is a contradiction. This completes the proof of Lemma 6.
3. The Proof of Theorem 1
Our proof is divided into the following three steps.
Step 1. Problem (1) has a weak solution in .
Let . In terms of Lemma 4, we know that is coercive in ; hence, is well defined. Ekeland’s variational principle guarantees to extract a minimizing sequence with
We assume that , in and (up to subsequence if necessary) converges to a nonnegative function, denoted by satisfying
Fix with , , by using Lemma 5, there exists a sequence of functions such that and for all and small enough. It follows from the definition of that so thus,
We derive that
Furthermore, there exists a constant such that . Taking , we deduce from Fatou’s lemma that
Consequently,
Since is arbitrary, this inequality also holds for ; thus, which implies that is a weak solution of the problem (1).
Step 2. There exists a constant such that when .
For each , we have
We only need to prove that , when with
Repeating the arguments as in Step 1, we have
In the following, we show that . Note that is closed and in , it suffices to prove that . We suppose that and satisfy the same properties as in (20) and (21). Let in the support of and . Define with
Applying (74), we derive that for every . Since , repeating the arguments as in Lemma 3, we derive that . Moreover,
Thus, by , namely, , we get or .
Next, we prove that does not hold. Otherwise, there exists such that , then where with ; this contradicts the fact . Choosing , we have and when .
Step 3. is a nontrivial solution to problem (1).
Let , then Lemmas 2–9 and Steps 1 and 2 hold for all . Using Lemma 3, we have
Hence, is a weak solution of (1). Since
Dividing by and passing to the limit as , we obtain
Hence, is a nontrivial solution of (1). This completes the proof of Theorem 1.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
This project is supported by the National Natural Science Fund (11326083).