- Open Access
On the sub-supersolution method for -Kirchhoff type equations
Journal of Inequalities and Applications volume 2012, Article number: 283 (2012)
This paper deals with the sub-supersolution method for the -Kirchhoff type equations. A sub-supersolution principle for the Dirichlet problems involving -Kirchhoff is established. A strong comparison theorem for the -Kirchhoff type equations is presented. We also give some applications of the abstract theorems obtained in this paper to the eigenvalue problems for the -Kirchhoff type equation.
MSC:35D05, 35D10, 35J60.
In this paper, we study the following problem:
where Ω is a bounded smooth domain in with , with , , is a continuous function with and satisfies the following condition:
() is a continuous and increasing function with .
The operator is said to be -Laplacian. The study of various mathematical problems with the variable exponent growth condition has received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. We refer the reader to  for an overview of and references on this subject.
The solvability of the problem (1.1) can be studied by several approaches; for example, the variational method (see, e.g., ). It is well known that, compared with other methods, the sub-supersolution method, or the order method, when it is applicable, has some distinctive advantages. For example, it usually gives some order properties of the solutions. For the applications of the sub-supersolution method to semilinear and quasilinear elliptic problems, we refer to [3, 4] and the references therein. In , Fan established a sub-supersolution principle for Dirichlet problems involving -Laplacian and a strong comparison theorem for -Laplacian equations. The goal of this paper is to study the sub-supersolution method for (1.1), which is a new research topic.
In , the sub-supersolution principle for -Laplacian equations established by Fan is based on the properties of -Laplace, the regularity results and the comparison principle. The aim of the present paper is to establish a sub-supersolution principle for -Kirchhoff equations.
The rest of this paper is organized as follows. In Section 2, we establish a general principle of the sub-supersolution method for the problem (1.1) based on the regularity results and the comparison principle. In Section 3, we give a special strong comparison principle for the -Kirchhoff. In Section 4, we give an application of our abstract theorems.
2 Sub-supersolution principle
In this section, we give a general principle of sub-supersolution method for the problem (1.1) based on the regularity results and the comparison principle. We would like to point out that the comparison principle in this section (see Theorem 2.2) is a generalization of Proposition 2.3 of . In addition to the principle of sub-supersolution, we shall establish also a generalization of Theorem 2.1 of . For simplicity, we write .
Definition 2.1 (1) We say that is a weak solution of (1.1) if
for any .
is called a subsolution (respectively a supersolution) of (1.1) if on ∂ Ω and, for all with ,
Regularity results and comparison principles are the basis of the sub-supersolution method. For the regularity results in the variable exponent case, see [7–9]. More precisely, for the and regularity, see ; for the local regularity of the minimizers of the corresponding integral functional, see ; for the global regularity, see .
If f is independent of u, we have
Theorem 2.1 If () holds and , , then (1.1) has a unique weak solution.
Proof Clearly, (for any ) defines a continuous linear functional on X. According to Theorem 4.1 of , is a homeomorphism. So, (1.1) has a unique solution, where . □
From Theorem 2.1 we know that, for a given , where and
has a unique solution under the condition (). We denote by the unique solution. K is called a solution operator for (2.2).
From the regularity results and the embedding theorems, we can obtain the properties of the solution operator K as follows.
Proposition 2.1 (1) If () holds, the mapping is continuous and bounded. Moreover, the mapping is completely continuous since the embedding is compact.
If () holds and p is log-Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
If () holds and p is Hölder continuous on , then the mapping is bounded, and hence the mapping is completely continuous.
Definition 2.2 Let . We say that if for all with ,
Now we give the comparison principle as follows.
Theorem 2.2 (1) Let and () hold. If and , then in Ω.
Under the conditions of (1) above, let in addition and denote . If S is a compact subset of Ω, then .
Proof (1) Taking in the proof of Theorem 3.2 of , we can get the conclusion.
Suppose that S is a compact subset of Ω and . Then there is an open subset of Ω such that . Thus on and consequently there is an such that on . Noting that and applying the conclusion (1) to u and on , we obtain in , which contradicts on S. □
It follows from Theorem 2.2(1) that the solution operator K is increasing under the condition (), that is, if . We define . It is easy to see that if u is a subsolution (respectively a supersolution) of (1.1), then (respectively ), and u is a solution of (1.1) if and only if , i.e., u is a fixed point of T.
The basic principle of the sub-supersolution method for (1.1) is the following result.
Theorem 2.3 Let () hold and suppose that f satisfies the sub-critical growth condition
and the function is nondecreasing in . If there exist a subsolution and a supersolution of (1.1) such that , then (1.1) has a minimal solution and a maximal solution in the order interval , i.e., and if u is any solution of (1.1) such that , then .
Proof Define . Then, under the assumptions of Theorem 2.3, is completely continuous and increasing, , , , , and consequently . It is clear that the cone of all nonnegative functions in is normal. Noting the minimal (maximal) fixed point (see ) of T is the minimal (maximal) solution of (1.1), so our Theorem 2.3 now follows by applying the well-known fixed point theorem for the increasing operator on the order interval (see, e.g., ). □
In the practical problems, it is often known that the subsolution and the supersolution are of class , so the restriction on the growth condition of f is needless. Hence, the following theorem is more suitable.
Theorem 2.4 Let () hold and suppose that , and are a subsolution and a supersolution of (1.1) respectively, and . If satisfies the condition
then the conclusion of Theorem 2.3 is valid.
The above results show that the general principle of the sub-supersolution method for -Kirchhoff type equations (1.1) is of the same type as in the case of -Laplacian type equations. An essential prerequisite for the sub-supersolution method is to find a subsolution and a supersolution such that . It is well known that the homogeneity of the p-Laplacian operator and the positivity of the first eigenvalue of p-Laplacian Dirichlet problem play an important role in finding sub- and supersolutions of the p-Laplacian equation . Unlike the p-Laplacian, when is not identical with a constant, the -Laplacian operator is inhomogeneous and usually the infimum of its eigenvalues is 0. It is obvious that the eigenvalues of (1.1) are , where and are, respectively, the eigenvalues and eigenfunctions of in X. Thus, usually, the infimum of is also 0. Therefore, it is often difficult to find a subsolution and a supersolution of (1.1) with .
At the end of this section, we give a lemma which is useful to find a supersolution of (1.1). We denote by the best embedding constant of .
Lemma 2.1 Let () hold, and let u be the unique solution of the problem
Set . Then, when , , and when , , where and are positive constants depending on , , N, , and .
Proof Let u be the solution of (2.4), Theorem 2.2 implies . For , set . Taking as a test function in (2.4) and using the Young inequality, we have
When , taking
Consequently, from this and (2.5), it follows that
From (2.5) and (2.6), we have
By Lemma 5.1 in [, Chapter 2], (2.7) implies that
where . From (2.7) and (2.8), we obtain
When , taking
(noting that in this case ) and using arguments similar to those above, we can obtain
The proof is complete. □
Remark 2.1 We would like to point out that the fact that a solution of (2.4) is bounded in is useful for finding a supersolution of (1.1). Indeed, the fact can be used to estimate the relation of nonlinearity and ℳ (for details, see the proof of Theorem 4.1).
3 A strong comparison principle for -Kirchhoff problem
The energy functional associated with the problem (1.1) is
where and . In this section, we give a special strong comparison principle for the -Kirchhoff, which is suitable for finding a positive local minimizer of the integral functional J in the topology. In , Fan established a Brezis-Nirenberg type theorem (Theorem 1.1 of ), which asserts that every local minimizer of J in the topology is also a local minimizer of J in the topology. Applying this theorem, we have the following special form.
Theorem 3.1 Let (), (2.1) hold and let be a local minimizer (resp. a strictly local minimizer) of J in the topology. Then is a local minimizer (resp. a strictly local minimizer) of J in the X topology.
Applying Theorem 1.1 of , we can easily get the following strong maximum principle.
Theorem 3.2 Suppose that , , and in Ω. If
where , , , with , then in Ω.
Now we give a special strong comparison principle for the -Kirchhoff.
Theorem 3.3 Let () hold and suppose that , in Ω, ,
and in Ω. If
where n is the inward unit normal on ∂ Ω, then in Ω and there is a positive constant ε such that
Proof We denote by the inward unit normal at . For , set . Denoting , as in the proof of , we have
We claim that in . Indeed, if in , then in , and consequently in . Take such that in Ω, on . By (3.1) and the property of φ, we have
which contradicts . Hence the claim is true. So, by the well-known strong maximum principle for linear elliptic equations, in and (3.2) holds. Setting , then S is a compact subset of Ω. By Theorem 2.2(2), , hence in Ω and the proof is complete. □
The following theorem provides a method to find a positive local minimizer of the integral functional J in the topology.
Theorem 3.4 Let () hold and suppose that are a subsolution and a supersolution of (1.1) respectively, , , , , and in Ω. Suppose that , satisfies the condition of Theorem 2.3. If neither nor is a solution of (1.1), or neither nor is a minimizer of J on in the case of being a solution of (1.1), then there exists such that , is a solution of (1.1) and is a local minimizer of J in the topology.
As an application of the above abstract theorems, let us consider the following eigenvalue problem:
where Ω is a bounded smooth domain in , , , , , for and , is nondecreasing in , is fixed. The energy functional associated with the problem (4.1) is
Firstly, we recall the (PS) c condition and the mountain pass lemma which we shall use later.
Definition 4.1 Let X be a Banach space. We say that I satisfies the (PS) c condition in X if any sequence , such that and as , has a convergent subsequence, where (PS) means Palais-Smale.
Lemma 4.1 (see )
Let X be a Banach space, , and be such that and
If φ satisfies the (PS) c condition with
then c is a critical value of φ.
The main results are the following.
Theorem 4.1 Suppose that f satisfies the condition either
in Ω, or
and there are an open set , a closed ball , some positive constants and such that for and , and for .
Then we have the following assertions:
For sufficiently small , (4.1) has a solution which is a local minimizer of in the topology. Moreover, as .
Define and . Then and Λ are both intervals, and .
In addition, suppose that , for and
where for and . Then for each , (4.1) has at least two solutions and such that and is a local minimizer of in the topology.
Proof (1) Take , where h is as in Lemma 2.1, and let be the unique positive solution of (2.4). Then by Lemma 2.1, . Because , we can choose ℳ small enough such that , which implies that . Let be sufficiently small such that . Then for such λ,
which shows that v is a supersolution of (4.1) and is not a solution of (4.1). By Theorem 3.2, in Ω and on ∂ Ω.
In the case when f satisfies the condition (i), 0 is a subsolution of (4.1) and 0 does not satisfy the equation in (4.1). Moreover, by Theorem 3.4, (4.1) has a solution , which is a local minimizer of in the topology.
In the case when f satisfies the condition (ii), 0 satisfies the equation in (4.1). We claim that 0 is not a minimizer of on . To see this, noting , it is sufficient to show that . For , denote . By the condition (ii), we can find sufficiently small positive constants ρ such that , . Define a function such that and on . Then for sufficiently small , we have that and
which shows that the claim is true. By Theorem 3.4, there exists such that , is a solution of (4.1) and is a local minimizer of in the topology.
When , we can take , consequently and . Furthermore, and . Assertion (1) is proved.
Note that, under additional assumptions, it is easy to verify that and satisfies the (PS) c condition for all λ. Now let be given arbitrarily. Take with , and let , and be the solutions of (4.1), (4.1 λ ) and (4.1) respectively, , and let be a local minimizer of in the topology. Then by Theorem 3.1, is also a local minimizer of in the topology. Define
Consider the problem
and denote the associated functional to (4.2) by . It is easy to see that and are a subsolution and a supersolution of (4.2), respectively. By Theorem 3.4, there exists such that is a solution of (4.2) and is a local minimizer of in the topology. By Theorem 2.2(1), we can see that and consequently is also a solution of (4.1 λ ). If , then assertion (3) already holds, hence we can assume that . Now is a local minimizer of in the topology, and so also in the topology. We can assume that is a strictly local minimizer of in the topology, otherwise we have obtained assertion (3). It is easy to verify that, under the additional assumptions in the statement (3), and satisfies the (PS) c condition. From , () and , it follows that . Using Lemma 4.1, we know that (4.2) has a solution such that , as a solution of (4.2), must satisfy , and consequently, by Theorem 3.2 and Theorem 3.3, . Noting that is also a solution of (4.1 λ ) since , thus the proof of assertion (3) is complete. □
Note that in the case of Theorem 4.1(1) and (2), the variational method cannot be used directly because we do not suppose that and do not restrict the growth rate of f.
Diening L, Harjulehto P, Hästö P, Ruzicka M Lecture Notes in Mathematics 2017. In Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Berlin; 2011.
Fan XL:On nonlocal -Laplacian Dirichlet problems. Nonlinear Anal. 2010, 72: 3314–3323. 10.1016/j.na.2009.12.012
Fan XL:On the sub-supersolution methods for -Laplacian equations. J. Math. Anal. Appl. 2007, 330: 665–682. 10.1016/j.jmaa.2006.07.093
Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 1976, 18: 620–709. 10.1137/1018114
Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.
Dai G, Ma R:Solutions for a -Kirchhoff type equation with Neumann boundary data. Nonlinear Anal., Real World Appl. 2011, 12: 2666–2680. 10.1016/j.nonrwa.2011.03.013
Acerbi E, Mingione G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal. 2001, 156: 121–140. 10.1007/s002050100117
Fan XL, Zhao D: A class of De Giorgi type and Hölder continuity. Nonlinear Anal. 1996, 36: 295–318.
Fan XL:Global regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 2007, 235: 397–417. 10.1016/j.jde.2007.01.008
Dai, G: Three solutions for a nonlocal Dirichlet boundary value problem involving the -Laplacian. Appl. Anal., 1–20 (2011) (iFirst). doi:10.1080/00036811.2011.602633
Ma R, Dai G, Gao C:Existence and multiplicity of positive solutions for a class of -Kirchhoff type equations. Bound. Value Probl. 2012., 2012: Article ID 16
Drábek P, Hernández J: Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Anal. 2001, 44: 189–204. 10.1016/S0362-546X(99)00258-8
Ladyzenskaja OA, Ural’tzeva NN: Linear and Quasilinear Elliptic Equations. Academic Press, New York; 1968.
Fan XL:A Brezis-Nirenberg type theorem on local minimizers for -Kirchhoff Dirichlet problems and applications. Differ. Equ. Appl. 2010, 2(4):537–551.
Fan XL, Zhao YZ, Zhang QH:A strong maximum principle for -Laplace equations. Chin. J. Contemp. Math. 2003, 24(3):277–282.
Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.
Research was supported by the NSFC (No. 11261052, No. 11101335, No. 11061030).
The authors declare that they have no competing interests.
GD conceived of the study, and participated in its design and coordination and helped to draft the manuscript. XH participated in the design of the study. All authors read and approved the final manuscript.
About this article
Cite this article
Han, X., Dai, G. On the sub-supersolution method for -Kirchhoff type equations. J Inequal Appl 2012, 283 (2012). https://doi.org/10.1186/1029-242X-2012-283
- nonlocal problems
- comparison theorem