Abstract
In this paper, we study the nonlocal Laplacian problem of the following form
By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence and multiplicity of solutions of this problem. Moreover, we get much better results with f in a special form.
MSC: 35B38, 35D05, 35J20.
Keywords:
critical points; Laplacian; nonlocal problem; variable exponent Sobolev spaces1 Introduction
In this paper, we consider the following problem:
where is a function defined on , is a continuous function, satisfies the Caratheodory condition.
The operator is called Laplacian, which becomes pLaplacian when (a constant). The Laplacian possesses more complicated nonlinearities than pLaplacian; for example, pLaplacian is homogeneous, that is, for every ; but the Laplacian operator, when is not a constant, is not homogeneous. These problems with variable exponent are interesting in applications and raise many difficult mathematical problems. Some of the models leading to these problems of this type are the models of motion of electrorheological fluids, the mathematical models of stationary thermorheological viscous flows of nonNewtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. We refer the reader to [17] for the study of Laplacian equations and the corresponding variational problems.
Kirchhoff has investigated the equation
which is called the Kirchhoff equation. This equation is an extension of the classical d’Alembert’s wave equation by considering the effect of the changes in the length of the string during vibrations. A distinguishing feature of the Kirchhoff equation is that the equation contains a nonlocal coefficient which depends on the average of the kinetic energy on . Various equations of Kirchhoff type have been studied by many authors, especially after the work of Lions [8], where a functional analysis framework for the problem was proposed; see, e.g., [924] for some interesting results and further references. And now the study of a nonlocal elliptic problem has already been extended to the case involving the pLaplacian; see, e.g., [25,26]. Corrêa and Figueiredo in [16] present several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the pKirchhoff type equation. Recently, the Kirchhoff type equation involving the Laplacian of the form
has been investigated by Autuori, Pucci and Salvatori [27]. In [28] Fan studied Kirchhoff type equations with Dirichlet boundary value problems. Many papers are about these problems in bounded domains. According to the information I have, for Kirchhofftype problems in , the results are seldom, in [29] Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhofftype problems in , and in [30] Ji established the existence of infinitely many radially symmetric solutions of Kirchhofftype Laplacian equations in . The main difficulty here arises from the lack of compactness. Jin [29] and Ji [30] investigated these problems in radial symmetric spaces. In this paper, to deal with problem (P), we overcome the difficulty caused by the absence of compactness through the method of weight function. We establish conditions ensuring the existence and multiplicity of solutions for the problem.
This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces. In Section 3, we obtain the solutions with negative energy by the coercivity of functionals, and in Section 4, we obtain the solutions with positive energy by the Mountain Pass Theorem. Finally in Section 5, we obtain the infinity of solutions by the Fountain Theorem and the Dual Fountain Theorem when f satisfies a special form.
2 Preliminaries
In order to discuss problem (P), we need some theories on space which we call variable exponent Sobolev space. Firstly, we state some basic properties of space which will be used later (for details, see [6,31,32]).
Let Ω be an open domain of , denote by the set of all measurable real functions defined on Ω, elements in which are equal to each other and almost everywhere are considered as one element, and denote
we can introduce the norm on by
and becomes a Banach space. We call it a variable exponent Lebesgue space.
and it can be equipped with the norm
where ; and we denote by the closure of in , , , when , and , when .
Proposition 2.1 (see [6] and [31])
(1) If, the spaceis a separable, uniform convex Banach space, and its dual space is, where. For anyand, we have
Proposition 2.2 (see [6])
Ifis a Caratheodory function and satisfies
where, , andis a constant, then the superposition operator fromtodefined byis a continuous and bounded operator.
Proposition 2.3 (see [6])
If we denote
then forProposition 2.4 (see [6])
If, , then the following statements are equivalent to each other
Proposition 2.5 (see [6])
(1) If, thenandare separable reflexive Banach spaces.
Proposition 2.6Ifis Lipschitz continuous and, then forwith, there is a continuous embedding.
For any measurable functions α, β, use the symbol to denote
Proposition 2.7Let Ω be a bounded domain in, , . Then for anywith, there is a compact embedding.
Proposition 2.8 (Poincare inequality)
There is a constant, such that
3 Solutions with negative energy
In the following sections, we consider problem (P), the nonlocal Laplacian problem with variational form, where M is a real function satisfying the following condition: (M_{1}) = is continuous and bounded.. And we assume that , is Lipschitz continuous, , satisfies Caratheodory conditions.
For simplicity, we write . Denote by C a general positive constant (the exact value may change from line to line).
Before giving our main results, we first give several lemmas that will be used later.
Let (M_{1}) hold. Then the following statements hold:
Lemma 3.2 (see [2])
Suppose
where, , , , , and there aresuch that
Thenand Φ, are weaklystrongly continuous, i.e., impliesand.
Lemma 3.3
(1) The functionalis sequentially weakly lower semicontinuous, is sequentially weakly continuous, and thusEis sequentially weakly lower semicontinuous.
(2) For any open setwith, the mappingsandare bounded, and are of type, namely,
Proof Since the function is increasing and the functional is sequentially weakly lower semicontinuous, we conclude that the functional is sequentially weakly lower semicontinuous. From Lemma 3.2, we know that and are sequentially weaklystrongly continuous. Now let . It is clear that the mapping and are bounded. To prove that is of type , assuming that , in X and , then there exist positive constants and such that . Noting that . It follows from that , where . Since is of type . Moreover, since is sequentially weaklystrongly continuous, the mapping is of type . □
Definition 3.1 Let . A functional satisfies condition if and only if every sequence in X such that , and in has a convergent subsequence.
Lemma 3.4 (see [28])
Supposefsatisfies the hypotheses in Lemma 3.2, and let (M_{1}) hold. Then, for any, every boundedsequence forE, i.e., a bounded sequencesuch thatand, has a strongly convergent subsequence.
As X is a separable and reflexive Banach space, there exist and such that
Lemma 3.5 (see [2])
Assume thatis weaklystrongly continuous and, is a given positive number. Set
Theorem 3.1Supposefsatisfies the hypotheses in Lemma 3.2, let (M_{1}) hold and the following conditions hold: (M_{2}) = There are positive constants, MandCsuch thatfor.; (H_{1}) = ..Then the functionalEis coercive and attains its infimum inXat some. Therefore, is a solution of (P) ifEis differentiable at, and in particular, if.
Proof We have concluded that E is weakly lower semicontinuous. Let us prove that E is coercive on X, i.e., as . For simplicity, we assume that and denote , , , . We have that
and hence E is coercive. Since E is sequentially weakly lower semicontinuous and X is reflexive, E attains its infimum in X at some . In the case where E is differentiable at , is a solution of (P). □
Theorem 3.2Supposefsatisfies the hypotheses in Lemma 3.2. Let (M_{1}), (M_{2}), (H_{1}) and the following conditions hold: (M_{3}) = There is a positive constantsuch that.; (f_{1}) = There exists a positive constant,
where, , , , .; (H_{2}) = ..Then (P) has at least one nontrivial solution which is a global minimizer of the energy functionalE.
Proof From Theorem 3.1 we know that E has a global minimizer . It is clear that and consequently . As and , we can find a bounded open set such that for . The space is a subspace of X. Take . Then, by (f_{1}), (M_{3}) and (H_{2}), for sufficiently small , we have that
Theorem 3.3Let the hypotheses of Theorem 3.2 hold, andfsatisfy the following condition: (f_{2}) = forand..Then (P) has a sequence of solutionssuch that, andas.
Proof Denote by the genus of A. Denote
From the condition on , there exists a bounded open set such that for . The space is a subspace of X. For any k, we can choose a kdimensional linear subspace of such that . As the norms on are equivalent to each other, there exists such that with implies . is compact, and then there exists a constant such that
As , we can find and such that , , which implies , . Since , we get the conclusion .
By the genus theory, each is a critical value of E, hence there is a sequence of solutions of problem (P) such that .
At last, we will prove as . By the coercive of E, there exists a constant such that when . For any , let and be the subspace of X as mentioned above. According to the properties of genus, we know that . Set
we know as . When and , we have , and then , which concludes as . □
Theorem 3.4Let the hypotheses of Lemma 3.2, (f_{1}), (M_{1}), (M_{2}), (M_{3}), (H_{1}), (H_{2}) and the following condition hold, (f_{+}) = forand..Then (P) has at least one nontrivial nonnegative solution with negative energy.
Proof Define
Then, like in the proof of Theorem 3.2, using truncation functions above, similarly to the proof of Theorem 3.4 in [28], we can prove that has a nontrivial global minimizer and is a nontrivial nonnegative solution of (P). □
4 Solution with positive energy
In this section we will find the Mountain Pass type critical points of the energy functional E associated with problem (P).
Lemma 4.1Let (f_{1}), (M_{1}) and the following conditions hold: (M_{2})^{′} = , , andsuch that
withhold.; (M_{4}) = , such that
; (f_{3}) = , such that
; (H_{3}) = ..Then E satisfies conditionfor any.
Proof By (M_{4}), for large enough, we have
By (f_{3}) we conclude that there exists such that
and thus, given any , there exists such that
we claim that there exists such that
the notation of this conclusion can be seen in [28].
Now let , and . By (H_{3}), there exists small enough such that . Then, since is a sequence, for sufficiently large n, we have
we conclude that is bounded, since . By Lemma 3.4, E satisfies condition for . □
Lemma 4.2Under the hypotheses of Lemma 4.1, for any, as.
Proof Let be given. From (M_{4}) for sufficiently large we have
and then it follows that
where is a positive constant depending on w. From (f_{4}) for large enough we have
which implies that
where is a positive constant depending on w. Hence for s large enough, we have
Lemma 4.3Under the hypotheses of Lemma 3.2, (M_{1}) holds and the following conditions hold: (M_{5}) = There is a positive constantsuch that.; (f_{4}) = There existssuch thatforand
Then there exist positive constantsρandδsuch thatfor.
Proof It follows from (M_{5}) that
It follows from the hypotheses of Lemma 3.2 and (f_{4}) that
Thus by (H_{4}), we obtain the assertion of Lemma 4.3. □
By the famous Mountain Pass lemma, from Lemmas 4.14.3, we have the following:
Theorem 4.1Let all hypotheses of Lemmas 4.14.3 hold. Then (P) has a nontrivial solution with positive energy.
5 The case of concaveconvex nonlinearity
In this section, we will obtain much better results with f in a special form. We have the following theorem:
Then we have
(1) If (M_{1}), (M_{2})^{′}, (M_{4}), (H_{3}) hold and we also assume thatand, then problem (P) has solutionssuch thatas.
(2) If (M_{1}), (M_{4}), (M_{5}), (H_{3}) hold and we also assume thatand, then problem (P) has solutionssuch that, as.
We will use the following ‘Fountain Theorem’ and the ‘Dual Fountain Theorem’ to prove Theorem 5.1.
Proposition 5.1 (Fountain Theorem, see [11])
Assume(A_{1}) = Xis a Banach space, is an even functional, the subspaces, andare defined by (3.2)..
If for each , there existssuch that(A_{2}) = as.; (A_{3}) = .; (A_{4}) = Esatisfies thecondition for every. ThenEhas a sequence of critical values tending to +∞..
Proposition 5.2 (Dual Fountain Theorem, see [11])
Assume (A_{1}) is satisfied and there is aso as to for each, there existssuch that(B_{1}) = .; (B_{2}) = .; (B_{3}) = as.; (B_{4}) = Esatisfiescondition for every. ThenEhas a sequence of negative critical values converging to 0..
Definition 5.1 We say that E satisfies the condition (with respect to ), if any sequence such that , , and , contains a subsequence converging to a critical point of E.
Proof of Theorem 5.1 Firstly, we verify the condition for every . Suppose , , and . It is easy to obtain that satisfies condition (), when it has this special form. So similar to the method in Lemma 4.1, we have that
hence, we can get that is bounded. Going if necessary to a subspace, we can assume that in X. As , we can choose such that . Hence
As is of type, we can conclude ; furthermore, we have .
It only remains to prove . For any and we have
Going to the limit on the right side of the above equation reaches
so , this shows that E satisfies the condition for every . Obviously, E also satisfies the condition for every .
(1) We will prove that if k is large enough, then there exist such that (A_{2}) and (A_{3}) are satisfied. (A_{2}) For , denote
then , , and , , as . When , ,
For sufficiently large k, we have . As , we get
Since , we have . (A_{2}) is satisfied.
Then . For any , with and t large enough, since , all norms are equivalent in , we have
As , there exists such that concludes and then
so (A_{2}) is satisfied.
Conclusion (1) is reached by the Fountain Theorem.
(2) We use the Dual Fountain Theorem to prove conclusion (2), and now it remains for us to prove that there exist such that if k is large enough (B_{1}), (B_{2}) and (B_{3}) are satisfied.
(B_{1}) Let and be defined as above, when , and t small enough we have
For sufficiently large k we have , thus
Choose , then for sufficiently large k, . When , with , we have , which implies
Hence (B_{1}) is satisfied.
then . For , and t small enough, we have
with small enough. Hence (B_{2}) is satisfied.
(B_{3}) From the proof above and , we have
For , and small enough, we have
hence . Hence (B_{3}) is satisfied.
Conclusion (2) is reached by the Dual Fountain Theorem. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
EG and PZ contributed to each part of this work equally. All the authors read and approved the final manuscript.
Acknowledgements
The authors thank the two referees for their careful reading and helpful comments on the study. Research was supported by the National Natural Science Foundation of China (10971088), (10971087) and the Fundamental Research Funds for the Central Universities (lzujbky2012180).
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