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 spaces
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 p-Laplacian when (a constant). The -Laplacian possesses more complicated nonlinearities than p-Laplacian; for example, p-Laplacian 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 thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium. We refer the reader to [1-7] 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 , where a functional analysis framework for the problem was proposed; see, e.g., [9-24] 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 p-Laplacian; see, e.g., [25,26]. Corrêa and Figueiredo in  present several sufficient conditions for the existence of positive solutions to a class of nonlocal boundary value problems of the p-Kirchhoff type equation. Recently, the Kirchhoff type equation involving the -Laplacian of the form
has been investigated by Autuori, Pucci and Salvatori . In  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 Kirchhoff-type problems in , the results are seldom, in  Jin and Wu obtained three existence results of infinitely many radial solutions for Kirchhoff-type problems in , and in  Ji established the existence of infinitely many radially symmetric solutions of Kirchhoff-type -Laplacian equations in . The main difficulty here arises from the lack of compactness. Jin  and Ji  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.
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.
The space is defined by
and it can be equipped with the norm
where ; and we denote by the closure of in , , , when , and , when .
(1) If , the space is a separable, uniform convex Banach space, and its dual space is , where . For any and , we have
(2) If , then for any , , and ,
Proposition 2.2 (see )
If is a Caratheodory function and satisfies
where , , and is a constant, then the superposition operator from to defined by is a continuous and bounded operator.
Proposition 2.3 (see )
If we denote
(2) ; ;
(3) as ; as .
Proposition 2.4 (see )
If , , then the following statements are equivalent to each other
(3) in measure in Ω and .
Proposition 2.5 (see )
(1) If , then and are separable reflexive Banach spaces.
Proposition 2.6If is Lipschitz continuous and , then for with , 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 any with , there is a compact embedding .
Proposition 2.8 (Poincare inequality)
There is a constant , such that
So, is a norm equivalent to the norm in the space .
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: (M1) = 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).
Let , , define
Before giving our main results, we first give several lemmas that will be used later.
Let (M1) hold. Then the following statements hold:
(1) , , for .
(2) , , , and
Lemma 3.2 (see )
where , , , , , and there are such that
Then and Φ, are weakly-strongly continuous, i.e., implies and .
(1) The functional is sequentially weakly lower semi-continuous, is sequentially weakly continuous, and thusEis sequentially weakly lower semi-continuous.
(2) For any open set with , the mappings and are bounded, and are of type , namely,
Proof Since the function is increasing and the functional is sequentially weakly lower semi-continuous, we conclude that the functional is sequentially weakly lower semi-continuous. From Lemma 3.2, we know that and are sequentially weakly-strongly 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 weakly-strongly 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 )
Supposefsatisfies the hypotheses in Lemma 3.2, and let (M1) hold. Then, for any , every bounded sequence forE, i.e., a bounded sequence such that and , has a strongly convergent subsequence.
As X is a separable and reflexive Banach space, there exist and such that
For , denote
Lemma 3.5 (see )
Assume that is weakly-strongly continuous and , is a given positive number. Set
then as .
Theorem 3.1Supposefsatisfies the hypotheses in Lemma 3.2, let (M1) hold and the following conditions hold: (M2) = There are positive constants , MandCsuch that for .; (H1) = ..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 semi-continuous. Let us prove that E is coercive on X, i.e., as . For simplicity, we assume that and denote , , , . We have that
When is large enough, we have
and hence E is coercive. Since E is sequentially weakly lower semi-continuous 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 (M1), (M2), (H1) and the following conditions hold: (M3) = There is a positive constant such that .; (f1) = There exists a positive constant ,
where , , , , .; (H2) = ..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 (f1), (M3) and (H2), for sufficiently small , we have that
Hence which shows . □
Theorem 3.3Let the hypotheses of Theorem 3.2 hold, andfsatisfy the following condition: (f2) = for and ..Then (P) has a sequence of solutions such that , and as .
Proof Denote by the genus of A. Denote
we have .
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 k-dimensional 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
For and , we have
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, (f1), (M1), (M2), (M3), (H1), (H2) and the following condition hold, (f+) = for and ..Then (P) has at least one nontrivial nonnegative solution with negative energy.
Then, like in the proof of Theorem 3.2, using truncation functions above, similarly to the proof of Theorem 3.4 in , 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 (f1), (M1) and the following conditions hold: (M2)′ = , , and such that
with hold.; (M4) = , such that
Proof By (M4), for large enough, we have
By (f3) 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 .
Now let , and . By (H3), 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 (M4) for sufficiently large we have
and then it follows that
where is a positive constant depending on w. From (f4) for large enough we have
which implies that
where is a positive constant depending on w. Hence for s large enough, we have
and then as . □
Lemma 4.3Under the hypotheses of Lemma 3.2, (M1) holds and the following conditions hold: (M5) = There is a positive constant such that .; (f4) = There exists such that for and
uniformly in .; (H4) = ..
Then there exist positive constantsρandδsuch that for .
Proof It follows from (M5) that
It follows from the hypotheses of Lemma 3.2 and (f4) that
Thus by (H4), we obtain the assertion of Lemma 4.3. □
By the famous Mountain Pass lemma, from Lemmas 4.1-4.3, we have the following:
Theorem 4.1Let all hypotheses of Lemmas 4.1-4.3 hold. Then (P) has a nontrivial solution with positive energy.
5 The case of concave-convex nonlinearity
In this section, we will obtain much better results with f in a special form. We have the following theorem:
Theorem 5.1Let , where
(1) If (M1), (M2)′, (M4), (H3) hold and we also assume that and , then problem (P) has solutions such that as .
(2) If (M1), (M4), (M5), (H3) hold and we also assume that and , then problem (P) has solutions such 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 )
Assume(A1) = Xis a Banach space, is an even functional, the subspaces , and are defined by (3.2)..
If for each , there exists such that(A2) = as .; (A3) = .; (A4) = Esatisfies the condition for every . ThenEhas a sequence of critical values tending to +∞..
Proposition 5.2 (Dual Fountain Theorem, see )
Assume (A1) is satisfied and there is a so as to for each , there exists such that(B1) = .; (B2) = .; (B3) = as .; (B4) = Esatisfies condition 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 (A2) and (A3) are satisfied. (A2) For , denote
then , , and , , as . When , ,
For sufficiently large k, we have . As , we get
Choose , we have
Since , we have . (A2) is satisfied.
(A3) For , denote
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 (A2) 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 (B1), (B2) and (B3) are satisfied.
(B1) 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 (B1) is satisfied.
(B2) For , denote
then . For , and t small enough, we have
since and , we get
with small enough. Hence (B2) is satisfied.
(B3) From the proof above and , we have
For , and small enough, we have
hence . Hence (B3) is satisfied.
Conclusion (2) is reached by the Dual Fountain Theorem. □
The authors declare that they have no competing interests.
EG and PZ contributed to each part of this work equally. All the authors read and approved the final manuscript.
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 (lzujbky-2012-180).
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