Issue |
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 4, August 2024
|
|
---|---|---|
Page(s) | 365 - 373 | |
DOI | https://doi.org/10.1051/wujns/2024294365 | |
Published online | 04 September 2024 |
Mathematics
CLC number: O175.8
Unilateral Global Bifurcation and One-Sign Solutions for Kirchhoff Type Problem in ℝN
全空间ℝN上Kirchhoff型方程单侧全局分歧和保号解
College of General Education, Guangdong University of Science and Technology, Dongguan 523083, Guangdong, China
Received:
21
September
2023
In this paper, we study the following Kirchhoff type problem: Unilateral global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of for the existence, nonexistence, and exact multiplicity of one-sign solutions for this problem.
摘要
本文研究了下列Kirchhoff型方程: 建立了方程的单侧全局分歧结果。应用上述分歧结果, 对于属于不同区间的值, 得到了方程保号解的存在性,不存在性及解的确切个数。
Key words: unilateral global bifurcation / one-sign solutions / Kirchhoff type problem
关键字 : 单侧全局分歧 / 保号解 / Kirchhoff型方程
Cite this article: SHEN Wenguo. Unilateral Global Bifurcation and One-Sign Solutions for Kirchhoff Type Problem in [J]. Wuhan Univ J of Nat Sci, 2024, 29(4): 365-373.
Biography: SHEN Wenguo, male, Ph.D., Professor, research direction: nonlinear functional differential equations. E-mail: shenwg369@163.com
Fundation item: Supported by the National Natural Science Foundation of China (11561038)
© Wuhan University 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Consider the following semi-linear elliptic problem
where is a real parameter, , and for some is a weighted function which can be sign-changing and , and for any . Edelson et al[1,2] studied the existence of positive solution and the existence of global branches of minimal solutions of the problem (1) by the Schauder-Tychonoff fixed point theorem and Dancer global bifurcation theorems[3], respectively. By using Rabinowitz global bifurcation method[4], Rumbos et al[5] showed the existence of positive minimal solution of the problem (1). In 2017, Dai et al[6] established a global bifurcation result for problem (1).
On the other hand, Lions[7] studied the following Kirchhoff type problem
where is a bounded domain in with a smooth boundary . The problem (2) is nonlocal as the appearance of the term which implies that it is not a pointwise identity. By applying the bifurcation techniques, Liang et al[8] and Figueiredo et al[9] also studied equation (2). Dai et al[10] studied the problem (2) by Rabinowitz[4].
Motivated by the above papers, we shall study the following Kirchhoff type problem
where is a real parameter. By Ref. [6], set
For any with , we define . Denote by the completion of with respect to the norm . Denote by the set of all measurable real functions defined on . Two functions in are considered as the same element of when they are equal almost everywhere. Let
We assume that , and satisfy the following conditions:
(A1) Let . If there exist two continuous positive radially symmetric functions and , where (where and are given in (A3) and Section 1) such that and
Furthermore, if satisfies the following stronger condition (with )
(A2) is a Holder continuous function with exponent and
(A3) There exist and such that where
(A4) is increasing, , .
(A5) There exists , such that .
Furthermore, we shall investigate the existence of one-sign solutions for the following Kirchhoff type problems
We assume that satisfies (A1), and satisfies the following assumptions:
(H1) is a Holder continuous function with exponent such that for any .
(H2)
(H3) .
(H4)
(H5)
(H6)
(H7)
(H8)
(H9)
(H10)
where
Finally, we shall study the exact multiplicity of one-sign solutions for (5) by Implicit Function Theorem, the stability properties and condition (A6).
(A6) such that is decreasing in and is increasing in .
The rest of this paper is arranged as follows. In Section 1, we give some preliminaries and establish the unilateral global bifurcation result for the problem (3). In Section 2, on the above unilateral global bifurcation result, we prove the existence of one-sign solutions for the Kirchhoff type problem (5). In Section 3, we study the exact multiplicity of one-sign solutions for (5).
1 Preliminaries
Let with the norm . Let and set and .
Now, from Theorem 1.1 in Ref. [6], we know that the following eigenvalue problem
possesses a unique principal eigenvalue , and is simple and isolated.
To prove Theorem 1, by Section 4 of Ref. [6], we first consider the following problem
Let us define the operator by
where being the volume of the unit ball in .
Then by an argument similar to that of Ref. [1], we can show that is a one-sign solution of problem (7) if and only if is a solution of the operator equation . Similar to proposition 1 in Ref. [5], we also can show that is linear completely continuous and (8) is equivalent to (7).
The first main result for (3) is the following unilateral global bifurcation theorem.
Theorem 1 Assume that (A1)-(A5) hold. The pair is a bifurcation point of the problem (3) and there are two distinct unbounded continua and in of solutions of the problem (3) emanating from . Moreover, we have where .
Proof By p.5960-5961 in Ref. [6], it is clear that the problem (3) can be equivalently written as where
From conditions (A1)-(A5) and noting , we can see that is completely continuous. Furthermore, it follows that is completely continuous and
Next, we show at uniformly on bounded sets. Without loss of generality, we may assume that . Otherwise, we can consider such that . From , we can see So we can choose a real number such that
It follows
By (A2) and (A3), for any , we can choose positive numbers and such that the following relations hold:
Then we can obtain
By and the continuous embedding of , we have
Moreover, as , we obtain that
Let by the boundedness of , and the continuous embedding of , we have furthermore, we can get
We obtain
uniformly
By (10), we have uniformly for and on bounded sets, i.e. at uniformly on bounded sets.
Furthermore, applying the similar proof method of Theorem 1.3 in Ref. [6] and the Rabinowitz global bifurcation theorem[4], one can obtain that is a bifurcation point of the problem (3) and there exists one unbounded continua of solutions of the problem (3) emanating from .
Moreover, by the Dancer unilateral global bifurcation theorem[11], we have that there are two distinct unbounded bifurcation continua and in of solutions of the problem (3) emanating from . Moreover, we have
where .
2 One-Sign Solutions for Kirchhoff Type Problem
We first have the following results.
Remark 1 From (H1) and (H2), we can see that there exist two positive constants such that for all .
By an argument similar to that of Lemma 4.1, 4.2 in Ref. [6], we can obtain Lemma 1 and 2.
Lemma 1 Let (H1) and (H2) hold. By Remark 1, the problem (5) has no one-sign solution for any
Lemma 2 Let (H1) and (H2) hold. By Remark 1, the problem (5) has no positive solution for any The main results of this section are the following theorem.
Theorem 2 Let (A1), (A4), (A5), (H1) and (H2) hold. For any the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof Let be a continuous function such that , with , uniformly a.e. in . Equation (5) can be divided in the form
where
Using the same method to prove (10) with obvious changes, it follows that
Moreover, we have
uniformly on bounded sets.
By Theorem 1, there are two distinct unbounded continua and in of solutions of the problem (5) emanating from , such that
where .
We will show that joins to . Let
satisfy
By Remark 1 and Lemma 2, one can obtain for all . It follows from Lemma 1 that there exists a constant such that for any . Therefore, we get
One can get that joins to . Let be a continuous function such that , with
uniformly a.e. in . We divide the equation
where
By (12), for any , we can choose positive numbers and such that for a.e. ,the following relations hold:
Then we can obtain
By , we have
Moreover, as , we obtain
Let by the boundedness of , (9) and the continuous embedding of , we have
Furthermore, we can get
It follows from that
uniformly Furthermore, one obtain
uniformly for on bounded sets.
By the compactness of , we obtain
where , again choosing a subsequence and relabeling if necessary. Thus it is clear that since is closed in . Moreover, by (15), , so that . Thus joins to . Now the existence of and is clear.
Similar to the proof of the Theorem 1.3 in Ref. [6], we have
for some constants and .
Theorem 3 Let (A1), (A4), (A5), (H1) and (H3) hold. If the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof Inspired by the idea of Ref. [12], we define the cut-off function of as the following
We consider the following problem
Clearly, we can see that , and .
The Proposition 4.1 in Ref.[13] implies that there exist two sequence unbounded continua of solution set of problem (16) emanating from ,such that where .
Taking , we easily obtain that with . So condition (i) of Theorem 1.2 in Ref.[13] is satisfied with .
Define a mapping such that
It is easy to verify that is a homeomorphism and . Obviously, is a sequence of unbounded connected subsets in , so (ii) of the Theorem 1.2 in Ref. [13] holds. Since is completely continuous from , we have is pre-compact, and accordingly (iii) of the Theorem 1.2 in Ref.[13] holds. Therefore, by the Theorem 1.2 in Ref.[13], is unbounded closed connected of solutions of the problem (5) emanating from and by the Proposition 5.1 in Ref.[13], such that either is unbounded in the direction of or meets some point on .
From (H1) and (H3), we obtain that there exists a positive constant such that for any . So, Lemma 1 implies is bounded in the direction of .Hence, meets for some . From Theorem 2, we can obtain and , where . Now the desired conclusion is obvious.
Theorem 4 Let (A1), (A4), (A5), (H1) and (H4) hold. If then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof In view of Theorem 2, there are two distinct unbounded continua and in of solutions of the problem (5) emanating from , such that where .
We only need to show that joins to . We shall only prove the case since the proof for the other case is completely analogous.
Suppose on the contrary that there exists be a blow-up point and . Then there exists a sequence such that and . Let . Then should be the solutions of problem
Similar to the proof of (14), we can show
By the compactness of and (17), we obtain that for some convenient subsequence . This contradicts .
Similar to the proof of Theorem 2, we have
for some constants and .
Theorem 5 Let (A1), (A4), (A5), (H1) and (H5) hold. If then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof If is any nontrivial solution of problem (5), dividing problem (5) by and setting yields
Define
and
Evidently, problem (18) is equivalent to
It is obvious that is always the solution of problem (19). By simple computation, we can show that and .
Now, applying Theorem 3 and the inversion , we achieve the conclusion.
Theorem 6 Let (A1), (A4), (A5), (H1) and (H6) hold. If then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof Applying a similar method as the proof of Theorem 5 and the conclusion of Theorem 4, we can easily get the desired conclusion.
Theorem 7 Let (A1), (A4), (A5), (H1) and (H7) hold. If then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof Define
Clearly, we can see that , and .
Theorem 4 implies that there exists a sequence of unbounded components of solutions to problem (20) emanating from and joins to .
The Lemma 2.5 in Ref. [13] implies that there exists an unbounded component of such that and where
Theorem 8 Let (A1), (A4), (A5), (H1) and (H8) hold. If then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof Define
Clearly, we can see , and .
Theorem 3 implies that there exists a sequence of unbounded components of solutions to problem (21) emanating from and joins to .
The Corollary 2.1 in Ref. [13] implies that there exists an unbounded component of such that and , where
Theorem 9 Let (A1), (A4), (A5), (H1) and (H9) hold. There exists a , such that , then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof In view of Theorem 8, there are two distinct unbounded continua of solutions of the problem (5) emanating from . Similar to the proof of Theorem 4, we can obtain that joins to .
Theorem 10 Let (A1), (A4), (A5), (H1) and (H10) hold. There exists a such that then the problem (5) possesses two solutions , such that and in . Therefore, we have
for some constants and .
Proof In view of Theorem 7, there are two distinct unbounded continua of solutions of the problem (5) emanating from . Similar to the proof of Theorem 3, we can obtain that joins to .
3 Exact Multiplicity of One-Sign Solutions for Problem (5)
Refs. [14,15] studied exact multiplicity of solutions for a semi-linear elliptic equation, respectively.
In this section, we study exact multiplicity of one-sign solutions for problem (5). We first study the local structure of the bifurcation branch () near , which is obtained in Theorem 1. Let and
For and , define an open neighborhood of in as follows.
Let be a closed subset of satisfying , where is an eigenfunction corresponding to with . According to the Hahn-Banach theorem, we have satisfying where denotes the dual space of . For any and , define
Obviously, is an open subset of , , with which are disjoint and open in .
Similar to the Lemma 6.4.1 in Ref. [16], we can show the following lemma.
Lemma 3 Let , there is such that for each , it holds that
And there exist and a unique such that , for each .
Further, and as for these solutions .
Remark 2 From (H2) and (A6), we can see that for any , and .
Remark 3 From Lemma 3, we can see that near is given by a curve for s near . Moreover, we can distinguish between two portions of this curve by and .
Now, when , and satisfy the conditions (A1), (A4), (A5), (A6), by Dai et al[17] and Afrouzi et al[18], we give the definition of linearly stable solution for the problem (5) first.
For any and positive solution of problem (5), we can calculate that the linearized eigenvalue problem of (5) at the direction is
Definition 1 Suppose is a solution of problem (5). The linear stability of can be determined by the linearized eigenvalue problem (20). If all eigenvalues of problem (20) are positive, then we call is stable, otherwise we call it unstable.
The Morse index of is defined as the number of negative eigenvalues of problem (20). Call is degenerate if is an eigenvalue of problem (20), otherwise it is non-degenerate.
The main results of this paper are the following:
Theorem 11 Let (A1), (A4), (A5), (A6) and (H2) hold. If , then the problem (5) has exactly two solutions and such that and in , and has only the trivial solution for any
The following lemma is stability result for the positive solution.
Lemma 4 Under the assumptions of Theorem 11, then any solution of problem (5) is stable and non-degenerate, and their Morse index are .
Proof Let be a solution of problem (5), and let be the corresponding principal eigenpair of problem (20) with in . Notice that and satisfy
and
Multiplying the first equation of problem (22) by and the first equation of problem (21) by , subtracting and integrating, we obtain
By some simple computations, we can show that it follows from (A6) that for any . Since and in , we have and the positive solution must be stable. Similarly, we also have:
Lemma 5 Under the assumptions of Lemma 4, any negative solution of problem (5) is stable, hence, non-degenerate and Morse index .
Proof of Theorem 11 Define by
From Lemma 4 and Lemma 5, we know that any one sign solution of problem (5) is stable. Therefore, at any one-sign solution for the problem (5), we can apply the Implicit Function Theorem to , and all the solutions of near are on a curve with for some small . Furthermore, by virtue of Remark 3, the unbounded continua and are all curves.
To complete the proof, it suffices to show that is increasing (decreasing) with respect to . We only prove the case of . The proof of can be given similarly. Since is differentiable with respect to (as a consequence of Implicit Function Theorem), taking the derivative of the first equation of problem (21) by , one can obtain that
Multiplying the first equation of problem (23) by and the first equation of problem (21) by , subtracting and integrating, we obtain
Remark 2 implies for any . So we get by (A1). While (A6) shows that . Therefore, we have .
Next we only prove the case of the uniqueness of positive solution of problem (5) since the proof of the uniqueness of negative solution of problem (5) is similar.
Suppose on the contrary that there exist two solutions and corresponding to with of the problem (5) for . For , take then as . By the monotonicity of with respect to , we get Then .
By an argument as the above, we can show that problem (5) with has only the trivial solution. We can show that problem (5) has no one-sign solution for any . Suppose on the contrary that there exists a positive solution for the problem (5), we multiply the first equation of problem (21) by , and obtain after integrations by
where is a positive eigenfunction associated to . It follows that , which contradicts . Similar to the above proof, we can obtain that the problem (5) has no positive solution for any Furthermore, we can obtain that the problem (5) has only the trivial solution for any
Theorem 12 Let (A1), (A4), (A5), (A6) and (H4) hold. If , then the problem (5) has exactly two solutions and for such that and in , and has only the trivial solution for any
Proof By Theorem 4 and an argument similar to that of Theorem 11, we can prove it.
Theorem 13 Let (A1), (A4), (A5), (A6) and (H5) hold. If , then problem (5) has exactly two solutions and for such that and in , and has only the trivial solution for any .
Proof By Theorem 5 and an argument similar to that of Theorem 11, we can obtain it.
Theorem 14 Let (A1), (A4), (A5), (A6) and (H7) hold. If then the problem (5) has exactly two solutions and for such that and in .
Proof By Theorem 7 and an argument similar to that of Theorem 11, we can prove it.
References
- Edelson A L, Rumbos A J. Linear and semilikear eigenvalue problems in ℝN[J]. Communications in Partial Differential Equations, 1993, 18(1/2): 215-240. [CrossRef] [MathSciNet] [Google Scholar]
- Edelson A L, Furi M. Global solution branches for semilinear equations in ℝN[J]. Nonlinear Analysis: Theory, Methods & Applications, 1997, 28(9): 1521-1532. [CrossRef] [MathSciNet] [Google Scholar]
- Dancer E N. Global solution branches for positive mappings[J]. Archive for Rational Mechanics and Analysis, 1973, 52(2): 181-192. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
- Rabinowitz P H. Some global results for nonlinear eigenvalue problems[J]. Journal of Functional Analysis, 1971, 7(3): 487-513. [CrossRef] [MathSciNet] [Google Scholar]
- Edelson A L, Rumbos A J. Bifurcation properties of semilinear elliptic equations in ℝN[J]. Differential and Integral Equations, 1994, 7(2): 399-410. [CrossRef] [MathSciNet] [Google Scholar]
- Dai G W, Yao J H, Li F Q. Spectrum and bifurcation for semilinear elliptic problems in ℝN[J]. Journal of Differential Equations, 2017, 263(9): 5939-5967. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
- Lions J L. On some questions in boundary value problems of mathematical physics[M]//Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Eyuations. Amsterdam: Elsevier, 1978: 284-346. [CrossRef] [Google Scholar]
- Liang Z P, Li F Y, Shi J P. Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior[J]. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2014, 31(1): 155-167. [CrossRef] [MathSciNet] [Google Scholar]
- Figueiredo G M, Morales-Rodrigo C, Santos J RJr, et al. Study of a nonlinear Kirchhoff equation with non-homogeneous material[J]. Journal of Mathematical Analysis and Applications, 2014, 416(2): 597-608. [CrossRef] [MathSciNet] [Google Scholar]
- Dai G W, Wang H Y, Yang B X. Global bifurcation and positive solution for a class of fully nonlinear problems[J]. Computers & Mathematics with Applications, 2015, 69(8): 771-776. [CrossRef] [MathSciNet] [Google Scholar]
- Dancer E N, Phillips R. On the structure of solutions of non-linear eigenvalue problems[J]. Indiana University Mathematics Journal, 1974, 23(11): 1069-1076. [CrossRef] [Google Scholar]
- Ambrosetti A, Calahorrano R M, Dobarro F. Global branching for discontinuous problems[J]. Comment Math Univ Carolin, 1990, 31: 213-222. [MathSciNet] [Google Scholar]
- Dai G W. Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros[J]. American Institute of Mathematical Science, 2016, 36(10): 5323-5345. [MathSciNet] [Google Scholar]
- Shi J P, Wang J P. Morse indices and exact multiplicity of solutions to semilinear elliptic problems[J]. Proceedings of the American Mathematical Society, 1999, 127(12): 3685-3695. [CrossRef] [MathSciNet] [Google Scholar]
- Ouyang T C, Shi J P. Exact multiplicity of positive solutions for a class of semilinear problem, II[J]. Journal of Differential Equations, 1999, 158(1): 94-151. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
- Julián L G. Spectral Theory and Nonlinear Functional Analysis[M]. Boca Raton: Chapman and Hall/CRC, 2001. [Google Scholar]
- Dai G W, Han X L. Exact multiplicity of one-sign solutions for a class of quasilinear eigenvalue problems[J]. Journal of Mathematical Research with Applications, 2014, 34(1): 84-88. [MathSciNet] [Google Scholar]
- Afrouzi G A, Rasouli S H. Stability properties of non-negative solutions to a non-autonomous p-Laplacian equation[J]. Chaos, Solitons & Fractals, 2006, 29(5): 1095-1099. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.