On a Kirchhoff Equation in Bounded Domains
-
Yisheng Huang
Abstract
In this paper, we consider the following Kirchhoff equation:
where
1 Introduction
1.1 Background
In this paper, we consider the following elliptic equation:
where
The elliptic equation (Pabλ) is a so-called Kirchhoff-type equation, since it contains a nonlocal operator
where
Clearly, (Pabλ) has a natural variational structure in
where
We note that the study of finding sign-changing solution for elliptic equations is a hot topic in the area of nonlinear analysis, and a lot of excellent results have been obtained in recent years, see, for example, [3, 4, 6, 8, 28, 32, 34] and the references therein. However, to the best of our knowledge, except the references [10, 24, 36, 29, 39], there are few results devoted to the existence of sign-changing solutions of Kirchhoff-type equations. The main reason in our opinion is the difference between decompositions of the corresponding functionals of Kirchhoff-type equations and the related local problem. For example, in the case of
where
This different decomposition makes the study on the sign-changing solutions of Kirchhoff-type equations to be nontrivial, even though such study can be seen as a natural extension of that on the related local equations. Actually, by directly applying the results of methods dealing with local elliptic equations stated in the above references to (Pabλ), we have the following results:
(
Note that the local problem (
1.2 Sign-Changing Solutions
Let us state our first result on the existence and multiplicity of sign-changing solutions of (Pabλ).
Theorem 1.1.
Let
If
If
Remark 1.2.
(1) Theorem 1.1 actually extends the known result (R1) mentioned above to the case of
(2) Theorem 1.1 gives not only the existence and multiplicity results to the sign-changing solutions of (Pabλ) in general bounded domains but also an estimation of the number of nodal domains to these sign-changing solutions. It is worth to point out that, to the best of our knowledge, the number of nodal domains to the sign-changing solutions was only estimated in [29] for the least energy sign-changing solutions in general bounded domains and in [10] for the radial sign-changing solutions in a ball. Thus, Theorem 1.1 seems the first result on estimating the numbers of nodal domains for high energy sign-changing solutions in general bounded domains.
(3) In the case of
Indeed, suppose that u is a sign-changing solution of (Pabλ) which has at least
for all
This, together with the definition of
It follows that
which implies
1.3 Concentration Behavior and “Double” Energy
Since b is the parameter of the Kirchhoff-type nonlocal term in (Pabλ), and it is well known that the solutions of (Pabλ) are related to those of (
where
Then our results in these aspects can be stated as follows.
Theorem 1.3.
Let
For every
If
(1.3)We have
1.4 Radial Sign-Changing Solution
Let
Theorem 1.5.
Let
If
Remark 1.6.
(1) Theorem 1.5 (1) is just the known result (R2) mentioned before, which can be directly proved by using the method in [10]. Here, we actually give a different proof. Let us give some words about the differences here. Generally speaking, the argument in [10] is mainly consisted by the studies on two minimizing problems. The first one is
where
with
The second one is
where
In such an argument, one first needs to prove that the minimizing problem (1.4) can be attained by some
(2) Theorem 1.5 (2) is complete new to the best of our knowledge. Moreover, as we stated in Remark 1.2 (3), the existence of
(3) Due to the radial symmetry of the ball in
We re-denote the radial sign-changing solutions
As in Theorem 1.3, we can study the concentration behavior of the radial sign-changing solutions given by Theorem 1.5 and obtain the following results.
Theorem 1.7.
Let
For every
If
We have
Remark 1.8.
Compared with the concentration behaviors of the energy value obtained in [10], another novelty is Theorem 1.7 (3), which we give some precise estimates for these concentration behaviors. On the other hand, the concentration behaviors of the energy value in Theorem 1.7 are much better than that in Theorem 1.3 due to the radial symmetry of the ball and the mini-max descriptions of the energy values of radial sign-changing solutions found in Theorem 1.5.
1.5 Uniqueness of Radially Positive Solution
It is well known that the local problem (
Theorem 1.9.
Let
This paper is organized as follows. In Section 2, we will study the sign-changing solutions of (Pabλ) in a general bounded domain Ω and prove Theorems 1.1 and 1.3. In Section 3, we will study the radial sign-changing solutions of (Pabλ) in
Throughout this paper, for the sake of simplicity and clarity, we always denote
2 The Case of General Ω
2.1 An Auxiliary Functional
Let
for
where
for all
Let
Then it is easy to see that
where
It follows that all critical points of
Lemma 2.1.
Let
In particular, we have
Proof.
By the choice of
It follows that
From (2.3), it follows that
On the other hand, for all
It follows that
where
By (2.4), (2.5) and (2.7), we can easily see that
for
For every
Lemma 2.2.
Let
Proof.
Let
for
Since
Then, by
Thus, we must have
By Lemma 2.2, we can see that
Lemma 2.3.
Let
Proof.
Let c be a critical value of
On the other hand, by (2.4) and the definition of
From the definition of
By choosing
Let
where
Clearly, for every
Lemma 2.4.
Let
Proof.
Due to the semigroup property of
which, together with Lemma 2.1 and (2.8), implies
Thus, by choosing
Since
Lemma 2.5.
Let
Proof.
Let
where
there exists
If case (1) occurs, then, by Lemma 2.4 and the semigroup property of
Indeed, suppose the contrary. Since
Since
Since
where
If case (2.a) occurs, then, by choosing
This is a contradiction. Let us consider case (2.b) in what follows. Since
Therefore, we must have
Then π is even and continuous on
By the definitions, we can easily see that
Lemma 2.6.
Let
Proof.
By Lemma 2.3 and (2.11), we can see that
Thanks to [3, Proposition 4.2], we have
Lemma 2.7.
Let
Proof.
By Lemma 2.5, there exist
From Lemma 2.6, [3, Proposition 4.2] and the properties of the classical genus (cf. [30, Proposition 5.4]), it follows that
Now, by the definition of
Suppose that
which contradicts the definition of
Now, we can obtain the following proposition.
Proposition 2.8.
Let
Let
Since
Lemma 2.9.
Let
Proof.
For every
and
It follows that
for all
For every
Since
Lemma 2.10.
Let
Proof.
Take any
By the definition of
Note that
Let
Lemma 2.11.
Let
Proof.
We first borrow some ideas from [10, Lemma 2.1] to show that the conclusion holds for all
where
Define
Since
for all
Then, by a direct calculation, we have
and
Consider the matrix
where
and
Clearly,
for all
Note that
From the implicit function theorem once more, it follows that there exists only one branch bifurcated at 0, which is a contradiction. Thus, we must have
By claims 1 and 2,
Thus,
if
We next consider the case of
where
and
On the other hand, we also have
and
Thus, the matrix
From a similar argument as that used for the case of
Thus, by a similar argument as that used for the case of
Remark 2.12.
By checking the proof of Lemma 2.11, we can see that
Proposition 2.13.
Let
Proof.
Suppose on the contrary that
Thus,
which contradicts Lemma 2.10. For the sign-changing solution
2.2 Existence Results and Some Further Properties
Since the functional
if
Proposition 2.14.
Let
in the case of
Proof.
By the construction of
Since
we can see that (2.31) cannot hold when
Let us make some observations on the concentration behaviors of
Proposition 2.15.
Let
Proof.
We first consider the case of
for all
Thus, we actually have that
It follows that
Therefore,
By a standard argument, we can see that
Now, since
This is a contradiction. Now, by the above conclusion, it is easy to see that
Now, since
Proposition 2.16.
Let
Proof.
The conclusion for
It follows that
Thus,
Let us prove some more properties of the sign-changing solution of (Pabλ). We first give a technical lemma.
Lemma 2.17.
Let
where
Proof.
The proof is very similar to that of Lemma 2.11 for the case of
for all
for all
Let the sign-changing Nehari manifold of
Then it is easy to see that all sign-changing solutions of (Pabλ) are contained in
Proposition 2.18.
Let
Proof.
We only prove the case
By choosing
Now, thanks to Lemma 2.17once more, we can obtain
By Proposition 2.15, we can see that
Lemma 2.19.
Let
Proof.
We only give the proof for
and
where
Clearly, since
and
Thus, since
Since
Note that
and
which implies
Proposition 2.20.
Let
Moreover, if
Proof.
Since
It follows from the definition of
and
which, together with Lemma 2.19, implies
and
for almost
and
Thus, by (2.35) and (2.36), we can see that
Therefore,
where
We close this section by noting that the proof of Theorem 1.1 follows immediately from Propositions 2.13, 2.14 and 2.16, and the proof of Theorem 1.3 follows immediately from Propositions 2.15 and 2.18–2.20.
3 The Case of Ω = 𝔹 R
Since
Then by the symmetric criticality principle of Palais, the critical points of
3.1 An Auxiliary Elliptic System of Kirchhoff Type
For a positive integer
where
Let
Then it is easy to see that
Definition 3.1.
We say that
Let
Then it is easy to see that all nontrivial critical points of
Let
Lemma 3.2.
Let
Proof.
Let
Clearly,
Then, for every
and
for all
where
Thus, by
Lemma 3.3.
Let
Proof.
Consider the following system in
Since
and
for all
Then, for every
By Lemma 3.3, we can see that
Lemma 3.4.
Let
Proof.
Let
Multiplying this equation with
Let
and
for all
for all
and
for
due to
Let
It is easy to see that if
We set
where
By Lemma 2.11 once more, we can see that
Note that by a standard argument, we have that
where
Let
Then
Then, by Lemma 2.11 once more,
By Lemma 2.11 once more, we can see that
Lemma 3.5.
Let
Proof.
(1) Let
On the other hand, since
for all
which implies
Thus, by (3.9), we can see that
(2) By the construction of
Let
Lemma 3.6.
Let
Proof.
Assume that
Since
Lemma 3.7.
Let
Proof.
Let
Then, by the construction of
which implies that
Proposition 3.8.
Let
Proof.
By Lemma 3.7, we can easily see that
Since (3.1) is dependent on β, we re-denote
Lemma 3.9.
Let
and
Proof.
By a similar argument as that used for (3.9) in the case of
It follows that
for all
for all
Since
Note that
Hence,
3.2 Existence Results and Some Further Properties
Since
where
Lemma 3.10.
Let
Proof.
Let
and define
It follows that
By Lemma 3.10, without loss of generality, we may assume that there exists
Lemma 3.11.
Let
Proof.
From Lemmas 3.5 and 3.10, we have immediately that
It follows that
Proposition 3.12.
Let
Proof.
Let
for all
where
Note that by the mean value theorem, we have
Thus, for
For
In other words, we always have that
Since
Thus, by Lemma 3.11, we must have
As for the general bounded domain Ω, let us prove some more properties of
Proposition 3.13.
Let
Proof.
Let v be a radial sign-changing critical point of
Proposition 3.14.
Let
Proof.
Thanks to Lemmas 3.5 and 3.11, we can obtain the conclusion by a similar argument as that used in the proof of Proposition 2.15. ∎
Proposition 3.15.
Let
Proof.
By Proposition 3.14, we can show that
By Proposition 3.14, we can see that
Proposition 3.16.
Let
Moreover, if
Proof.
Since
We close this section by noting that the proof of Theorem 1.5 follows immediately from Propositions 3.12–3.13, and the proof of Theorem 1.7 follows immediately from Propositions 3.14–3.16.
3.3 Uniqueness of the Positive Solution
We re-denote (Pabλ) and
and
Then by similar arguments as those used in [1, 36], we can show that there exists a radially positive solution
where
and
and
Then
Set
Then
In particular,
if and only if
Lemma 3.17.
Let
Proof.
For the sake of clarity, we divide the proof into three steps. Step 1: We prove that
Note that
where
Hence, we have from (3.19) that
which, together with (3.18), implies
Set
Then, by a similar argument as that used in Lemma 2.19 (see also [7, Lemma 3.1]),
which, together with (3.18) and the fact that
Thus,
By the fact that
It follows that
in the case of
This, together with Lemmas 2.11 and 2.17 once more, implies
Thus, we must have
We re-denote the positive solution of
Lemma 3.18.
We have that
where
Proof.
The proof is similar to that of Lemma 2.19. ∎
We close this section by proving Theorem 1.9.
Proof of Theorem 1.9.
Let
Note that both
due to the fact that
for some
This is a contradiction. Thus,
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11471235
Award Identifier / Grant number: 11626226
Award Identifier / Grant number: 11701554
Award Identifier / Grant number: 11771319
Funding source: Fundamental Research Funds for the Central Universities
Award Identifier / Grant number: 2017XKQY091
Funding statement: Y. Huang was supported by the National Natural Science Foundation of China (11471235). Y. Wu was supported by the Natural Science Foundation of China (11626226, 11701554, 11771319) and the Fundamental Research Funds for the Central Universities (2017XKQY091).
Acknowledgements
The authors thanks Professor Jiangquan Liu for his friendship, encouragement and enlightening discussions.
References
[1] C. O. Alves and G. M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in
[2] A. Azzollini, The elliptic Kirchhoff equation in
[3] T. Bartsch, Z. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), no. 1–2, 25–42. 10.1081/PDE-120028842Search in Google Scholar
[4] T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a p-Laplacian equation, Proc. Lond. Math. Soc. (3) 91 (2005), no. 1, 129–152. 10.1112/S0024611504015187Search in Google Scholar
[5] G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), no. 3, 289–306. 10.1016/0022-1236(86)90094-7Search in Google Scholar
[6] Z. Chen, C.-S. Lin and W. Zou, Sign-changing solutions and phase separation for an elliptic system with critical exponent, Comm. Partial Differential Equations 39 (2014), no. 10, 1827–1859. 10.1080/03605302.2014.908391Search in Google Scholar
[7] Z. Chen and W. Zou, On the Brezis–Nirenberg problem in a ball, Differential Integral Equations 25 (2012), no. 5–6, 527–542. 10.57262/die/1356012677Search in Google Scholar
[8] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations 52 (2015), no. 1–2, 423–467. 10.1007/s00526-014-0717-xSearch in Google Scholar
[9] B. Cheng, X. Wu and J. Liu, Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity, NoDEA Nonlinear Differential Equations Appl. 19 (2012), no. 5, 521–537. 10.1007/s00030-011-0141-2Search in Google Scholar
[10] Y. Deng, S. Peng and W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in
[11] G. M. Figueiredo, N. Ikoma and J. a. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal. 213 (2014), no. 3, 931–979. 10.1007/s00205-014-0747-8Search in Google Scholar
[12] Z. Guo, Ground states for Kirchhoff equations without compact condition, J. Differential Equations 259 (2015), no. 7, 2884–2902. 10.1016/j.jde.2015.04.005Search in Google Scholar
[13] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in
[14] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in
[15] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in
[16] Y. Huang, Z. Liu and Y. Wu, On a critical Kirchoff problem in high dimensions, preprint (2016), https://arxiv.org/abs/1605.06906v1. Search in Google Scholar
[17] Y. Huang, Z. Liu and Y. Wu, On finding solutions of a Kirchhoff type problem, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3019–3033. 10.1090/proc/12946Search in Google Scholar
[18] Y. Huang, T.-F. Wu and Y. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in
[19] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Search in Google Scholar
[20] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in
[21] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), no. 7, 2285–2294. 10.1016/j.jde.2012.05.017Search in Google Scholar
[22] Z. Liang, F. Li and J. Shi, Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 1, 155–167. 10.1016/j.anihpc.2013.01.006Search in Google Scholar
[23] J. Liu, J.-F. Liao and C.-L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in
[24] A. Mao and Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (2009), no. 3, 1275–1287. 10.1016/j.na.2008.02.011Search in Google Scholar
[25] E. Müller-Pfeiffer, On the number of nodal domains for eigenfunctions of elliptic differential operators, J. Lond. Math. Soc. (2) 31 (1985), no. 1, 91–100. 10.1112/jlms/s2-31.1.91Search in Google Scholar
[26] D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), no. 4, 1168–1193. 10.1016/j.jde.2014.05.002Search in Google Scholar
[27] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), no. 1, 246–255. 10.1016/j.jde.2005.03.006Search in Google Scholar
[28] M. Ramos, H. Tavares and W. Zou, A Bahri–Lions theorem revisited, Adv. Math. 222 (2009), no. 6, 2173–2195. 10.1016/j.aim.2009.07.013Search in Google Scholar
[29] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations 259 (2015), no. 4, 1256–1274. 10.1016/j.jde.2015.02.040Search in Google Scholar
[30] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2nd ed., Ergeb. Math. Grenzgeb. (3) 34, Springer, Berlin, 1996. Search in Google Scholar
[31] J. Sun and T.-F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations 256 (2014), no. 4, 1771–1792. 10.1016/j.jde.2013.12.006Search in Google Scholar
[32] S. Terracini and G. Verzini, Multipulse phases in k-mixtures of Bose–Einstein condensates, Arch. Ration. Mech. Anal. 194 (2009), no. 3, 717–741. 10.1007/s00205-008-0172-ySearch in Google Scholar
[33] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), no. 7, 2314–2351. 10.1016/j.jde.2012.05.023Search in Google Scholar
[34] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 (2008), no. 1, 83–106. 10.1007/s00205-008-0121-9Search in Google Scholar
[35] Y. Wu, On a K-component elliptic system with the Sobolev critical exponent in high dimensions: The repulsive case, Calc. Var. Partial Differential Equations 56 (2017), no. 5, Paper No. 151. 10.1007/s00526-017-1252-3Search in Google Scholar
[36] Y. Wu, Y. Huang and Z. Liu, On a Kirchhoff type problem in
[37] Y. Wu, Y. Huang and Z. Liu, Kirchhoff type problems with potential well and indefinite potential, Electron. J. Differential Equations 2016 (2016), Paper No. 178. Search in Google Scholar
[38] Y. Wu, T.-F. Wu and W. Zou, On a two-component Bose–Einstein condensate with steep potential wells, Ann. Mat. Pura Appl. (4) 196 (2017), no. 5, 1695–1737. 10.1007/s10231-017-0635-6Search in Google Scholar
[39] Z. Zhang and K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006), no. 2, 456–463. 10.1016/j.jmaa.2005.06.102Search in Google Scholar
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Articles in the same Issue
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- Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
- Prescribing Gaussian and Geodesic Curvature on Disks
- Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions
- Singularly Perturbed Fractional Schrödinger Equation Involving a General Critical Nonlinearity
- Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian
- Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces
- Polynomial Solutions of Equivariant Polynomial Abel Differential Equations
- Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities
- A Diffusion Equation with a Variable Reaction Order
- Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations
- Singularly Perturbed Fractional Schrödinger Equations with Critical Growth
- On a Kirchhoff Equation in Bounded Domains
Articles in the same Issue
- Frontmatter
- Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4
- Prescribing Gaussian and Geodesic Curvature on Disks
- Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions
- Singularly Perturbed Fractional Schrödinger Equation Involving a General Critical Nonlinearity
- Finite-Dimensionality and Determining Modes of the Global Attractor for 2D Boussinesq Equations with Fractional Laplacian
- Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces
- Polynomial Solutions of Equivariant Polynomial Abel Differential Equations
- Existence of Extremal Functions for Higher-Order Caffarelli–Kohn–Nirenberg Inequalities
- A Diffusion Equation with a Variable Reaction Order
- Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝn and n-Laplace Equations
- Singularly Perturbed Fractional Schrödinger Equations with Critical Growth
- On a Kirchhoff Equation in Bounded Domains
