Abstract
In this paper, we use the shooting method to study the solvability of the boundary value problem of differential equations with signchanging weight function:
where
1 Introduction and main result
In this paper, we are interested in the multiplicity of positive solutions for the boundary value problem:
where
Existence and multiplicity of positive solutions of (1.1) with a signchanging weight function have been extensively studied, see [4,5]. In [6], the authors established multiplicity results of positive solutions with Dirichlet boundary conditions in relation to the nodal behavior of the weight
where
Theorem A
(Theorem 1.1, [7]) Let
and the weight term
has least three positive solutions
A natural question that arises from the aforementioned quoted papers is whether the number of positive solutions to the problem (1.1) is related to the number of zeros of
where
The aim of the present paper is to show how the three solutions theorem in [7] generalizes in case we increase the number of zeros of
Let
Theorem 1.1
Let
2 Proof of main theorem
To prove our main theorem, we need some preliminary results.
In this section, we will find three positive solutions
where
First, studying problem (2.1) in the interval
Lemma 2.1
Let
Proof
Let
by the monotonicity of
By integrating (2.3) on
Lemma 2.2
Let
and
Proof
By integrating (2.2) on
and therefore,
We suppose
Furthermore, we have
when
which implies a contradiction.
Similarly, we suppose
Integrate on
which implies a contradiction, and Lemma 2.2 is proved.□
Lemma 2.3
Let
Proof
Let
so, for all
For any
We suppose that there exists
By integrating of (2.2) on
Furthermore, we obtain
which implies a contradiction.□
Second, we consider problem (2.1) in the interval
Lemma 2.4
Let
then
Proof
Suppose that
which implies a contradiction.□
Lemma 2.5
Let
Proof
Let
we have
By integrating of (2.6) on
Lemma 2.6
Let
and
Proof
Let
We suppose
we have
By integrating (2.8) on
in particular,
By integrating (2.7) twice on
When
which implies a contradiction.
And then, we suppose
contradiction and Lemma 2.6 is proved.□
Finally, we studying problem (2.1) in the interval
Lemma 2.7
Let
Lemma 2.8
Let
and
The proof process is completely similar to Lemmas 2.1 and 2.2, and it is omitted here.
Proof of Theorem 1.1
We show that problem (2.1) has at least three solutions through the following five steps.
Step 1. What needs to be explained is that
with the initial conditions
Step 2. In interval
We also have
According to the intermediate value theorem, there exists an interval
Furthermore, apply Lemma 2.3, there exists
Similarly, there exists an interval
Step 3. In interval
Thus, there exists an interval
Step 4. In interval
and fix
fix
Meanwhile, apply Lemma 2.4, we have
According to the continuous dependence of the solutions upon the initial data and the Intermediate Value Theorem, for
such that
and for all
Step 5. In these connected regions, using the forward shooting method and the backward shooting method respectively, we can obtain at least three solutions to problem (2.1). At the same time, it is also the solution of problem (1.2). This completes the proof.□
Finally, we point out that, even if, for the sake of simplicity, we only consider the case that

Funding information: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12161079) and Natural Science Foundation of Gansu Province (No. 20JR10RA086).

Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.
References
[1] R. Benguria, H. Brezis, and E. H. Lieb, The ThomasFermivon Weizsacker theory of atoms and molecules, Commun. Math. Phys. 79 (1981), no. 2, 167–180, https://doi.org/10.1007/BF01942059. Search in Google Scholar
[2] E. Sovrano and F. Zanolin, Indefinite weight nonlinear problems with Neumann boundary conditions, J. Math. Anal. Appl. 452 (2017), no. 9, 126–147, https://doi.org/10.1016/j.jmaa.2017.02.052. Search in Google Scholar
[3] K. Nakashima, The uniqueness of indefinite nonlinear diffusion problem in population genetics part I, J. Differential Equations 261 (2016), no. 11, 6233–6282. 10.1016/j.jde.2016.08.041Search in Google Scholar
[4] R. Y. Ma and X. L. Han, Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function, Appl. Math. Comput. 215 (2009), no. 3, 1077–1083, https://doi.org/10.1016/j.amc.2009.06.042. Search in Google Scholar
[5] A. Boscaggin, A note on a superlinear indefinite Neumann problem with multiple positive solutions, J. Math. Anal. Appl. 377 (2011), no. 1, 259–268, https://doi.org/10.1016/j.jmaa.2010.10.042. Search in Google Scholar
[6] M. Gaudenzi, P. Habets, and F. Zanolin, A sevenpositivesolutions theorem for a superlinear problem, Adv. Nonlinear Stud. 4 (2004), no. 2, 149–164, https://doi.org/10.1515/ans20040202. Search in Google Scholar
[7] G. Feltrin and E. Sovrano, Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal. 166 (2018), 87–101, https://doi.org/10.1016/j.na.2017.10.006. Search in Google Scholar
[8] D. Papini and F. Zanolin, Atopological approach to superlinear indefinite bounary value problems, Topol. Methods Nonlinear Anal. 15 (2000), 203–233, https://doi.org/10.12775/TMNA.2000.017. Search in Google Scholar
[9] D. G. Aronson and H. F. Weinberger, Nonlinear Diffusion in Population Genetics Combustion and Nerve Pulse Propagation, Springer, Berlin, 1975. 10.1007/BFb0070595Search in Google Scholar
[10] P. B. Baily, L. F. Shampine, and P. E. Waltman, Nonlinear Twopoint Boundary Value Problems, Academic Press, New York, 1968. Search in Google Scholar
[11] E. H. Lieb, ThomasFermi and related theories of atoms and molecules, Rev. Modern Phys. 53 (1982), no. 4, 263–301, https://doi.org/10.1007/354027056620. Search in Google Scholar
[12] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen. 7 (1937), no. 4, 355–369, DOI: https://doi.org/10.1111/j.14691809.1937.tb02153.x. 10.1111/j.14691809.1937.tb02153.xSearch in Google Scholar
[13] E. Sovrano and F. Zanolin, Indefinite weight nonlinear problems with Neumann boundary conditions, J. Math. Anal. Appl. 452 (2017), no. 1, 126–147, https://doi.org/10.1016/j.jmaa.2017.02.052. Search in Google Scholar
[14] A. Boscaggin and M. Garrione, Positive solutions to indefinite neumann problems when the weight has positive average, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5231–5244, https://doi.org/10.3934/dcds.2016028. Search in Google Scholar
[15] H. R. Quoirin and A. Surez, Positive solutions for some indefinite nonlinear eigenvalue elliptic problems with Robin boundary conditions, Nonlinear Anal. 114 (2015), 74–86, https://doi.org/10.1016/j.na.2014.11.005. Search in Google Scholar
[16] S. I. Pohozaev and A. Tesei, Existence and nonexistence of solutions of nonlinear Neumann problems, SIAM J. Math. Anal. 31 (1999), no. 1, 119–133, https://doi.org/10.1137/S0036141098334948. Search in Google Scholar
[17] D. Bonheure, J. M. Gomes, and P. Habets, Multiple positive solutions of superlinear elliptic problems with signchanging weights, J. Differ. Equ. 214 (2005), no. 1, 36–64, https://doi.org/10.1016/j.jde.2004.08.009. Search in Google Scholar
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