Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents
Available accessResearch articleFirst published online October 30, 2020
Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents
In this paper, we study the following generalized quasilinear Schrödinger equation
where , , , , has a positive global minimum, and has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.
This article is concerned with the following generalized quasilinear Schrödinger equation
where , , has a positive global minimum, and has a positive global maximum and g satisfies:
is even with for all , . Moreover, there exist some constants and such that
and
Mathematically, it is also a hot issue in nonlinear analysis to study the existence of solitary wave solutions for the following quasilinear Schrödinger equation
where , is a given potential, and are suitable functions. For various types of l, the quasilinear equation of the form (1.3) has been derived from models of several physical phenomenon. In particular, was used for the superfluid film [30,31] equation in fluid mechanics by Kurihara [30]. For more physical background, we can refer to [4,6,14,25,32,41,46,48] and references therein. In fact, many conclusions about the equation (1.3) with for some have been studied, see [38–40] and the references therein. However, to our knowledge, in the recent papers, the equation (1.3) with a general l has been considered by [16–19,35–37,52].
Set , where and u is a real function, (1.3) can be reduce to the corresponding equation of elliptic type (see [15]):
If we take
then (1.4) turns into quasilinear elliptic equations (see [52])
For (1.5), there are many papers (see [11,52–54]) on the existence of standing wave solutions. For example, in [16,17], Deng et al. proved the existence of nodal solutions via variational method. Afterwards, in [18,19], Deng et al. proved the existence of positive solutions with critical exponents, where critical exponents are and , respectively. Specifically speaking, in [19], the authors studied the following equations with critical exponents
By constructing the limit functional, the authors established the existence of positive solutions via concentration compactness principle with a certain g. By the same arguments as [19], Deng, Peng and Yan [18] consider the following critical generalized quasilinear problem
where g satisfied more stricter conditions than [19]. Very recently, in [35], Li et al. proved the existence of ground state solutions and geometrically distinct solutions via Nehari manifold method. In [36], the authors established the existence of a positive solution, a negative solution and infinitely many solutions via symmetric mountain theorem. Moreover, Chen, Tang and Cheng [11] proved the existence of ground state solutions for (1.1) via non-Nehari manifold method. With regard to generalized quasilinear Schrödinger equation of Kirchhoff type and generalized quasilinear Schrödinger–Maxwell system, we can refer to [12,13,33,59] and references therein. For related contributions to the study of local or nonlocal Schrödinger equations we refer to the recent papers [2,3,42,44].
For concentration behavior of solutions for (1.1), there is only one paper to study the following equation
where , satisfies the following condition:
Under the above condition on the potential , Li and Wu [37] studied the existence, multiplicity, and concentration of solutions by the Ljusternik–Schnirelmann theory. Next, if we set and in (1.6) and without critical growth nonlinearity term, then (1.6) reduces to the following equation
For problem (1.8) with local potential condition given by
where Ω is bound domain in , Wang and Zou [56] proved the existence and concentration of bound states solutions with -condition by variational technique with . Specifically speaking, firstly, the authors used the Penalized method developed by Pino and Felmer [45] to overcome the locality of V, and they worked on a suitable Orlicz space and established the asymptotic behavior of V as . At last, they proved the exponential decay phenomenon of bound state solutions. For , do Ó and Severo [22] proved the existence and concentration behavior of positive ground state solutions for (1.8) in , where is a small positive parameter and V is a positive uniformly continuous. They studied (1.8) by allowing to enjoy critical exponential growth. They mainly used a version of Trudinger–Moser inequality, a penalization technique and mountain-pass argument in nonstandard Orlicz space. Afterwards for problem (1.8) with potential condition (1.7), He, Qian and Zou [27] proved the existence and concentration of positive ground state solutions with -condition by employing the Nehari manifold method, the mountain pass theorem and compactness argument. Specifically speaking, the authors employed an argument developed by Liu et al. [38] to reformulate (1.8) to a new one whose associated functional and Gâteaux differentiable is well defined in a suitable Orlicz space. They used some classical arguments developed by Brezis and Nirenberg [7] to prove that the corresponding energy functional satisfies -condition.
On the one hand, if in (1.8), there are many papers on the existence and concentration of solutions. In [51], the authors assumed that and are uniformly continuous function with possessing at least one minimum and having at least one maximum, where . They proved the conclusions by employing mountain pass theorem in a Orlicz space. Subsequently, in [55], the authors considered the same potential function in [51] and proved the existence and concentration of standing wave solutions via Pohozaev manifold method, which is originated from [49]. For more papers, we can refer to [8,43] and references therein. On the other hand, if , in [28], under proper assumptions, He and Li obtained the existence and concentration phenomena of solutions for (1.8) and got the multiple solutions by employing the topology of the set where the potential attains its minimum and attains its maximum.
It is a natural problem if the existence and concentration behavior of positive ground state solutions of (1.1) as with critical exponent in the traditional working space ? To our knowledge, the existence and concentration behavior of the ground solutions to Eq. (1.1) with critical exponent in has not ever been considered by variational methods. It is worth pointing out that there are many difficulties in treating this class of generalized quasilinear Schrödinger equation in . To this end, we need to overcome those difficulties:
lack of compactness;
the term prevents us from working directly in a classical working space;
the estimate of mountain pass level.
In this paper, we prove the existence of positive ground state solutions via Nehari manifold method for enough small , and we establish the concentration behavior and the exponential decay phenomenon of ground state solutions. Finally, by Ljusternik–Schnirelmann theory, we also obtain the existence of multiple solutions. Our results extend and supplement the results obtained by Deng–Peng–Yan [18,19] and He–Li [28] and some other related literatures. Before stating our results, we need to give some notations on V and K:
To state our results, we need to give the following assumptions on V and K.
, and are uniformly continuous in , ;
and there exists , such that for ;
and there exists , such that for .
By the condition (), we can assume , and let
If () holds, we assume , and let
It is obvious that and are bounded sets. Moreover, if , then . In particular, if K is a constant and if V is a constant.
Now, let us recall some basic notions. Let
with the norm
It is well known that in the study of the elliptic equations, the potential function V plays an important role in choosing a right working space and some suitable compactness methods. Generally speaking, there many papers to study (1.1) by using the following working space:
endowed with the norm
In such a case, we can deduce formally that the Euler-Lagrange functional associated with the equation (1.1) is
To resolve the equation (1.1), due to the appearance of the nonlocal term
the right working space seems to be
But it is easy to see that is not a linear space under the assumption of (g). To overcome this difficulty, for any , Shen and Wang [52] make a change of variable as
then
Therefore, by this change of variables X can be used as the working space and can be transformed into
Since g is a nondecreasing positive function, we get . From this and our hypotheses, it is clear that is well defined in .
Furthermore, one can easily derive that if is a critical point of (1.9), then is a classical solution to the equation (1.1). To obtain a critical point of (1.9), we only need to seek for the weak solution to the following equation
Here, we say that is a weak solution to the equation (1.10) if
Then it is standard to obtain that is a weak solution to the equation (1.10) if and only if v is a critical point of the functional in . To sum up, it is sufficient to find a critical point of the functional in to achieve a classical solution to the equation (1.1). Moreover, it is easy to check that .
Now, we state the first result by the following theorem.
Suppose that (g), () and () holds, then for sufficiently smalland eitherandorand:
Equation (
1.1
) has a positive ground state solution;
has a global maximum pointsuch that;
there exist two constantsandsuch that
Next, we state the second result by the following theorem.
Suppose that (g),and () holds, and we replaceby, then all conclusions of Theorem
1.1
are still true.
To obtain the multiplicity of weak solutions for (1.1), we need to give the following condition on and :
.
Before stating our results, we need to recall the notion about Ljusternik–Schnirelmann category: if Y is a closed set of a topological space X, the least number of closed and contractible sets in X which cover Y is called Ljusternik–Schnirelmann category of Y in X, denoted by . Let
be a closed σ-neighborhood of .
Now, the following result is given.
Suppose that (g), (), () (or ()) and () holds. Then for any given, there existssuch that for any, (
1.1
) has at leastsolutions if eitherandorand. Moreover, ifdenotes one of these solutions andpossesses a maximum, then
;
there exist two constantsandsuch that
There are indeed functions which satisfy (g). Some examples are given by:
and
Note that if we choose in (1.1), then (1.1) will become as the classical quasilinear Schrödinger equation
By Remark 1.4, we know that our results extend the paper [28]. Moreover, since we achieve the existence of ground state solutions in this paper, our results are different from [19] and it is a supplement.
To overcome the above difficulties, we need to explore more properties on g in this paper. By those properties, we give some estimates in the whole process of proof.
The paper is organized as follows. In Section 2, we give some preliminary lemmas, which play important roles in proving our results. In Section 3, we will prove Theorem 1.1 and Theorem 1.2 via Nehari manifold method. In Section 4, we will use Ljusternik–Schnirelmann theory to prove Theorem 1.3.
Throughout this paper, we make use of the following notations:
denotes and C possibly denotes the different constants in different space;
denotes the Lebesgue space with norm
for any and , ;
with the norm
the weak convergence in is denoted by ⇀, and the strong convergence by →;
denotes the exponential function.
denotes the set of all locally -integrable functions.
Some preliminary lemmas
In this section, we present some useful lemmas and corollaries. Now, let us prove the following lemma.
For the function g, G, and, the following properties hold:
the functionsandare strictly increasing and odd;
for all;
for all;
;
;
is increasing andfor all;
is non-decreasing inandfor all;
andfor all;
and;
and there exist some constantsandsuch thatwhere;
is decreasing in;
is non-decreasing infor;
for all.
()–() and () are a subsequence results of [24]. By (g), we have that for all ,
Integral on both sides of the above inequality, we get
which implies that
Moreover, if , by mean value theorem and the monotonicity of g, one has
If , the proof is analogous. This shows () holds.
Let
By L’Hospital rule, we have
which shows that . Thus .
In addition, by L’Hospital rule, we get . As to item (), it has been proved in [19].
For item (), let
then by (), we have
Thus () holds. Finally, let
then by (), we have
Thus () holds. □
Next, we make the change of variables, which reduces (1.1) to the following form
where
and
Define , (1.1) is equivalent to the following equation
Now, we give the following space. Define
endowed with the norm
Since V is bounded and , is equivalent to the standard norm in . Moreover, is continuous embedding for and thus there exists a constant such that
The energy functional of (2.1) is given by
There holdsuniformly for all,.
It follows from Lemma 2.1() and () that
and
which due to (). □
Next, we will show that the functional possesses the mountain pass geometry
There existboth independent of ε such thatfor allwith.
By Lemma 2.2, we have that for any , there exists such that
Thus by (2.2), one has
Taking and small enough, then there exists a constant such that . This completes the proof. □
There existswithsuch that.
By Lemma 2.2, we have that for any , there exists such that
Thus by (2.3), one has
Taking , it follows that as . Hence let with sufficiently large, then . This completes the proof. □
By Lemma 2.3 and Lemma 2.4, we know that possesses the mountain pass geometry. By mountain pass theorem in [47], we define the mountain pass level by
where
In order to obtain ground state solutions, we need to define the following Nehari manifold
Next, we give an important lemma, which play an important role in using Nehari manifold method.
For any, there exists a uniquesuch that. Moreover,.
Let . For , we define
Since if only if . Hence it follows from that
Note that the right hand of equality is an increasing function on . In fact, for any fixed , we consider a function given by
Let . Thus by () in Lemma 2.1, we know that is increasing in . Moreover let
then by () in Lemma 2.1, we can infer that is increasing in . Moreover, set
From () in Lemma 2.1 and (1.2), we have
and
Thus
Therefore by these three facts, we have that is increasing in .
Using Lemma 2.2 and (2.2), one has
by choosing . It follows that for small . Moreover, and by the above inequality, we know that for large. Therefore by the monotony of , we have that there exists a unique such that , that is, . Furthermore . This completes the proof. □
To obtain a ground state solutions, we need a characterization of the least energy. We consider the following
and
Next, we consider the following autonomous problem
where
The energy functional of (2.4) is given by
where
Similar to the above proof in Lemmas 2.3 and 2.4, it is easy to check that possesses the mountain pass structure and hence has a bounded -sequence, and its least energy has the same characterization as stated in Lemma 2.6. Using the fact that is invariant under translation, we know that is attained, where is the mountain pass level value and is given by
Moreover, the best Sobolev constant is defined by
is critical value ofifwhere S is given in (
2.5
).
By mountain pass theorem, we have that there exists a Cerami sequence in at level , that is,
Thus for any , we have
where as .
Let in (2.6). Then
and
which implies that
From (2.6), we have
and then
It follows from that we can conclude that is bounded. In fact, if as , then , which contradicts to . Thus there exists such that and . Next, we prove that is bounded in . To this end, we only need to show that is bounded in . Now, we prove it. Indeed, we spit
Hence by (2.5) and () in Lemma 2.1, one has
and
Thus is bounded in . Therefore there exists such that in .
Next, we claim that the vanishing does not hold, that is, there exist a sequence , and such that
If
Then by Lemma 1.21 in [57], we have
Similar to the proof in Lemma 2.2, we can get the following
For any , there exists a constant such that
where . Thus we have
Using , we have that
Let
Thus we have that . Moreover, by , it follows that
which implies that
Using the definition of S, we have
Thus . Together with (2.9), we get . This is a contradiction. Thus (2.7) holds.
Letting in (2.7), then
Since is bounded in . Then is bounded in . Up to a subsequence, there exists such that
and ϑ is a weak solution of (2.4).
By Fatou’s Lemma, we have
which implies that . This completes the proof. □
Now, we introduce a well-known fact (see [7]) that the minimization problem
has a solution given by
Let be a smooth cut-off function such that for , for and for , where , . Set , we can get the following estimations (see [1,9]).
Next, we can get the following useful lemma and the ideas of the proof is derived from [21].
There exist two positive constantsindependent of ϵ such that for ϵ small enough and
From Lemma 2.1(), for any given , there exists such that
For all , since , we have
which dues to and . Thus for ϵ small enough, we have . This together with (2.10), we get the result. □
If eitherandorand, then.
Firstly, we claim that for any small enough, there exists a unique such that
and
where and are positive constants independent of ϵ.
In fact, since , for sufficiently small and , there exists such that
Thus we have
By (2.8) and Lemma 2.8, for any , there exists a constant such that
This shows that
Thus
Moreover, by (2.11), we have
Thus we have
Next, we shall estimate . It follows from Lemma 2.8 that
Set
Then has only maximum at . Thus we have
In the following, we estimate
By Lemma 2.1 and Lemma 2.9, we have for small that
for small enough . On the one hand, one has
It follows from , that
Thus we have
From the above discussion and for small enough , we know
Moreover, we have
By direct calculation, we find if
then we can choose such that
and if
then there exists , such that
It follows from the above that for small enough. This completes the proof. □
Letand,, withandfor eitherandorand. Then. In particular, if one of the above inequality is strict, then.
By Lemma 2.7, we know that is attained. Let be a reach element of . Then
which implies that
Moreover let be such that
then we get
The second part of this lemma is similar to the argument in the first part of this lemma. The proof is completed. □
For anyand, up to subsequence, there holds
In this lemma, we shall use the same idea in [28]. Let , where ξ is a smooth cut-off function with , on , on , and ν is ground state solution to the equation which corresponding to the energy functional of . Thus there exists a unique such that
that is,
Similar to the proof in Lemma 2.10, we know that there exist two constants such that . Thus there exists such that as . Next, let , we have that
Letting in (2.12), we conclude that
Since , we get
From (2.13) and (2.14), one has
By Lemma 2.1() and (), we know that . Applying Lebesgue dominated convergence theorem, we can easy verify that
and
as . Thus
This completes the proof. □
Letbe such thatandwith. Then one of the following conclusions holds:
in;
there exist a sequence, R andsuch that
Suppose that (ii) is false. Then for all , up to a subsequence, we know that
By the vanishing lemma in [57], we have that in for all . Similar to the proof in (2.8), we can easily check that
uniformly for any and . Since . Thus one has
Moreover, by the definition of S, we have
If , it follows that
Since as , we can conclude that
By Lemma 2.9, for either and or and , we know that . This is a contradiction. Thus , and so (i) holds. This completes the proof. □
Assume thatbe bounded such thatandwithin. Ifin, then.
Assume that be a sequence such that . Next, we claim that the sequence satisfies . Assuming the contrary that there exist and a subsequence still denote by such that
Since is bounded in . Thus . It follows that
Noting that , we know
It follows from (2.15) and (2.16) that
For any , there exists such that
for all .
Since , and in for any , and together with Lemma 2.1() and (), we deduce that
and
which implies that
Since in , it follows from Lemma 2.13 that there exists and such that
If we set , then there exists a function such that, up to subsequence,
Moreover, by (2.18), there exists a subset with positive measure such that a.e. in Ω. It follows from that
for any . Taking the limit in the last inequality and applying Fatou’s Lemma, we have
which is a contradiction. Thus .
Next, we claim that as . In fact, if , by the fact that , for any , we have
Taking , one has
which implies that . By the fact that is bounded in , we can get a contradiction. Thus . Since and , using the mean value theorem, we know
From (2.15) and (2.19), one has
If , we have
By the assumption , in and Lemma 2.13, there exists with , such that in . Applying Fatou’s Lemma to (2.20), we have
which is a contradiction. Thus there exists a subsequence, denoted by such that as . Thus
Since is bounded in . Hence we have
Moreover, by the mean value theorem, one has
and
From (2.21)–(2.24), we deduce that
Letting in the last inequality, we have for any , which implies . □
satisfies the Cerami condition at any level.
Let such that and . By the same argument in the proof of Lemma 2.10, we know that is bounded in . Thus there exists such that in . Moreover, v is critical point of . Let . Similar to the proof in [20], we get
and for any ,
By () in Lemma 2.1, we have
Then . It follows from Lemma 2.14 that in , that is, in . □
In this section, we will prove Theorem 1.1 and Theorem 1.2. To this end, motivated by [28], we need to consider the following equation
where () satisfies
and is a Carathedory function such that for any , there exists such that
By the proof of Theorem 1.1 in [34], we have the following lemma.
Assume thatare weak solutions for (
3.1
) satisfyingfor. Ifis uniformly integrable near ∞, that is,,, for any,, then
Letbe a ground state solution to Eq. (
2.1
), for any maximum pointof, there holds.
For any sequence as , by Lemma 2.12 and Lemma 2.1(), we have
By Lemma 2.7, is bounded in . Similar to the proof of Lemma 2.13, there exist a sequence and , such that
Set , then satisfies the following equation
Since is bounded in , we can infer that is bounded in . Thus there exists such that
By the boundedness of and with and , up to a subsequence, there exist two constants such that
From the uniform continuity of V in , it is easy to check that as uniformly on any compact set. Similarly, uniformly on any compact set. Hence by (3.2), for any , we have
which implies that is a weak solution of the following equation
Moreover, one has
Thus .
Next, we claim that is bounded. Assume the contrary that , then and and then , which is a contradiction. Up to a subsequence, we may assume that , and .
Now, we will show . In fact, if , we know
which implies that . By the definition of , . Thus .
If , , just suppose that , then , which is a contradiction.
By Lemma 2.12, . Replace by in (3.3), we get
From Sobolev’s inequality and uniformly integrable near infinity, it follows from Lemma 3.1 that
Now we claim that if is a maximum point , then there exists such that
for all i. Indeed, by , it follows from (3.2) that
By Lemma 2.1() and (), we know that is increasing in . Then we have
Thus it follows that
which shows that
By (3.4), is bounded. Denote , it is clear that is a maximum point of . By the arbitrariness of , we complete the proof. □
Finally, we give an exponential decay of ground state solutions. Let be a ground state solution of the following equation
Set
It is easy to check that
Thus we can choose such that for all ,
and
Now, we define
where ζ and M are such that and for all ,
It is easy to check that for all ,
Thus
Let . Then it follows from (3.5), (3.6) and (3.7) that
By the maximum principle, we know that for all . Hence
which implies that
Since () holds. We assume that such that
By and , thus . By Lemma 2.7, Lemma 2.15 and the discussion of exponential decay, the proof is completed. □
The proof is similar to Theorem 1.1 and thus we omit it. The proof is completed. □
In this section, we study the multiplicity of solutions for Equation (1.1) by the Ljusternik–Schnirelmann category theory.
Suppose that Z is a Banach space, is a -manifold of Z and is a -functional. We say that satisfies -condition if any sequence such that and contains a convergent subsequence. Here denotes the norm of the derivative of restricted to at the point .
By Lemmas 2.7 and 2.10, we may take a fixed such that and . Let η be a smooth nonincreasing cut-off function defined in with , on , on , . For any , let
Then we can get the following lemma.
For any,and, there existssuch thatand.
By the definition of -norm, we have
Moreover,
as . Then for enough small . Therefore, by Lemma 2.5, there exists such that and
and
This completes the proof. □
For any , define by
, uniformly for.
If we suppose to the contrary, then there exist , and such that
Next, we claim that there exist two constants such that
In fact, by the definition of , we get
which implies that there exists such that . Since , there exists a positive constant such that
On the other hand, one has
which implies that there exists such that . Up to a subsequence, we can assume that there exists such that as . Next, similar to the proof in Lemma 2.12, we can complete the proof. □
For , let be such that . Consider the mapping defined by for and for . Moreover, let be
, uniformly for.
Let , one has
Since
Then by the fact that and is compact, we have
as , uniformly for . This completes the proof. □
Letbe a sequence satisfyingThen one of the following conclusions holds:
has a strongly convergent subsequence in;
there exists a sequencesuch that the sequenceconverges strongly in.
In particular, there exists a minimizer of c.
Let be such that
By () in Lemma 2.1, one has
By Lemma 2.7, is bounded in . Up to a subsequence, there exists such that
It is a standard to prove that for any . By the density of in , we get . In the following, we divide the proof into the following two cases.
Case I: if , then by the semi-continuity of norm, we get
Next, we claim that the above equality holds. If we assume that
then we have
which is a contradiction. Thus
Similar to the above proof, we have
Hence there exists such that
Thus by Lebesgue Dominated Converge theorem, we have
Moreover, there exists such that
Thus by Lebesgue Dominated Converge theorem, we have
On the one hand, we have
On the other hand, we have
Thus from the above inequality, we have
Hence in . Moreover, is a minimizer of c.
Case II: if , then we only need to prove that there exist and such that
Suppose to the contrary, we have
Applying the same arguments in Lemma 2.13, we can get
Since
and for any , there exists such that
Then , which contradicts . So letting , we get and . Thus it means that is a -sequence of . So there exists such that in . Next, the proof is the same as Case I. The proof is completed. □
Letandbe a sequence satisfyingThen there exists a sequencesuch that the sequencehas a convergent subsequence in. Moreover, up to a subsequence,.
By () in Lemma 2.1, one has
By Lemma 2.7, is bounded in . Consequently, by the proof of Lemma 2.13, we know that there exist a sequence and constants such that
Let . Up to a subsequence, there is a such that in . Let such that and . By the definition of and , it follows that
which shows that .
Next, we claim that . In fact, it is obvious that is bounded in . By the fact that in , then there exists a constant such that . So . It shows that is bounded and thus there exists such that . If , by the boundedness of , we get in , then , which is a contradiction. Consequently, . Let , , then
In fact, by Ekeland’s variational principle in [23], there exists a sequence satisfying
By the same argument as in Lemma 4.4, we have that
It follows from (4.1) and (4.2) that
and is a nontrivial critical point of . Then
Thus when , we have
and
Next, Similar to the proof in Lemma 4.4 of Case I, we have
Moreover by , we get
Next, we prove that . First, similar to the proof in Lemma 3.2, we can prove that is bounded and . Hence it suffices to show that and . Arguing by contradiction again, we assume that and . Since
For any compact set , there exists a compact set with such that for all small and all , . Since V is continuous and as , we have that
Since and , then is uniformly integrable near ∞, i.e. for any , there exists such that for all
By (4.5), (4.6), (4.7) and the boundedness of V, we have
Similarly,
Moreover, by (4.4), we have
which is impossible. Thus and . The proof is completed. □
Define the set
where for any . It follows from Lemma 4.2 that as . By the definition of , we know that, for any and , and .
Similar to the proof in [26–29,50], we can get the following lemma.
For any
Given , it follows from Lemma 4.2, Lemma 4.3 and Lemma 4.6 that there exists such that for any ,
is well defined. In view of Lemma 4.3, for ε small enough, we can denote by for , where uniformly for . Let . Thus is continuous. Obviously, and for all . It means that is homotopically equivalent to . By Lemma 4.3 of [5], we have
Since and as , we can use the definition of and Lemma 4.4 to conclude that satisfies the -condition in for all small . Therefore, by Chapter II, 3.2 of [10], we can prove that there are at least critical points of restricted to . Similar to the arguments in the proof of Lemma 4.4, we can know that a critical point of the functional on , which is a critical point of the functional in and therefore, up to a change of variables, it is a weak solution for the problem (1.1). Proceeding as we prove Theorem 1.1, we can complete the proof. □
Footnotes
Acknowledgements
The author thanks Professor Tang Xianhua for his thoughtful guidance. This work was supported by National Natural Science Foundation of China (Grant Nos. 11661053, 11801574, 11771198, 11901276, 11901345 and 11961045), and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009, 2018BAB201003), the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), Yunnan Local Colleges Applied Basic Research Projects (No. 2017FH001-011), Natural Science Foundation of Hunan Province (No. 2019JJ50788) and Central South University Innovation-Driven Project for Young Scholars (No. 2019CX022).
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