 Research
 Open Access
 Published:
Positive radial solutions for a noncooperative resonant nuclear reactor model with signchanging nonlinearities
Journal of Inequalities and Applications volume 2021, Article number: 74 (2021)
Abstract
This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system:
where \(\mathbb{R}^{N}\) (\(N\geq 1\)) is the usual Euclidean space, n indicates the outward unit normal vector, \(f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})\), \(g\in C([0,\infty ),[0,\infty ))\), and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature.
Introduction
Let \(N\geq 1\) be an integer, and let \(\Omega = \{x\in \mathbb{R}^{N}: R_{1}<x<R_{2}, 0<R_{1}<R_{2}< \infty \}\) be an annulus with boundary ∂Ω. In this paper, we establish the existence of positive radial solutions to the elliptic system
where n denotes the outward unit normal vector on ∂Ω, and c and d are positive constants. For convenience, we write \(q\gg 0\) for some function \(q\in C[R_{1},R_{2}]\) if it is strictly positive on \([R_{1},R_{2}]\), and we denote by q̄ and \(\underline{q}\) the maximum and minimum of \(q\gg 0\), respectively. Throughout the paper, we assume the following:

(H1)
\(f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})\), and there is \(\chi \gg 0\) such that
$$ p(t)f(t,u)\geq \chi (t)u, \quad (t,u)\in [R_{1},R_{2}] \times [0, \infty ), $$where \(p(t)=t^{N1}\), \(t\in [R_{1},R_{2}]\).

(H2)
\(g\in C([0,\infty ),[0,\infty ))\).
Obviously, the nonlinear term f is allowed to change its sign. Since the Laplace operator −Δ is not invertible under the Neumann boundary conditions, elliptic system (1.1) is resonant.
Elliptic system (1.1) is closely related to the stationary version of the mathematical model of nuclear reactors
where \(\Omega _{0}\subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary \(\partial \Omega _{0}\) and represents a closed container, u and v are respectively the density of the neutron flux and temperature of the nuclear reactors. \(b\in [0,\infty )\) and \(c,d\in (0,\infty )\) are constants, and \(u_{0}\) and \(v_{0}\) are continuous functions on \(\overline{\Omega }_{0}\). System (1.2) improves the original model
put forward in [1] by adding the diffusion and linear feedback of the temperature, where the Neumann boundary condition
means that the neutron flux cannot cross the boundary of the closed container, and the boundary of the closed container is heat insulation.
Over the past few decades, existence and related properties of positive stationary solutions of (1.3) (and its more general forms) have been studied by many authors; see Kastenberg and Chambré [1], Pao [2, 3], Gu and Wang [4], Arioli [5], LópezGómez [6], and the references therein. Meanwhile, some authors have also focused on the existence of positive solutions of the onedimensional analogue of (1.3). See, for instance, Wang and An [7–9], Li [10], Chen [11, 12], and references therein. However, as far as we know, most of papers mentioned are devoted to system (1.3) subject to Dirichlet boundary condition, which means that there is no neutron flux on the boundary of the container and the constant temperature on it, whereas the results associated with (1.4) are relatively rare. In addition, the existence results on positive solutions, obtained in [7–9, 11, 12], largely depend on the positivity of the nonlinearities, and only the nonresonant case has been treated. Based these reasons, our aim in the present paper is establishing the existence of positive radial solutions for elliptic system (1.1) at resonance.
To state our main results, we define
uniformly for \(t\in [R_{1},R_{2}]\).
Theorem 1.1
Assume (H1) and (H2). If \(g_{0}=0\), \(f_{\infty }=\infty \), and
then (1.1) has at least one positive radial solution.
Theorem 1.2
Assume (H1) and
 (H2)′:

\(g\in C([0,\infty ),[0,\infty ))\), and \(\lim_{u\to +\infty }p(t)g(u)=0\) uniformly for \(t\in [R_{1},R_{2}]\).
then (1.1) admits at least one positive radial solution.
Remark 1.1
(H1) implies that the nonlinearity f may be signchanging, and hence it is more general than the corresponding conditions in the existing literature. For the first time, we establish the existence results of elliptic system (1.1) in the resonant case; related results for other problems with sighchanging nonlinearities can be found in [13, 14] and references therein. To look for radially symmetric positive solutions, we impose a radial dependence of the coefficients involved in f, which is far from being the case in [15, 16] and most of the references therein; the results of these references can be adapted to deal with homogeneous Neumann boundary conditions, which we will do in some future work.
The rest of the paper is arranged as follows. In Sect. 2, we introduce some notations and preliminaries. In Sect. 3, we prove the main and some related results and give some remarks to demonstrate the feasibility of our main findings.
Preliminaries
As is well known, in finding a radial solution \((u, v)=(u(r), v(r))\), elliptic system (1.1) is equivalent to
where \(r=x\). Let \(t=r\) and \(p(t)=t^{N1}\). Then we have \(p(t)>0\) on \([R_{1},R_{2}]\), and the above system becomes
Hence, if we show that there is a positive solution to (2.1), then system (1.1) admits a positive radial solution. Here the positivity of a solution \((u,v)\) of (2.1) means that \(u, v\gg 0\).
Let us denote by \(K(t,s)\) the Green’s function of
Then it is easy to show that \(K(t,s)>0\) on \([R_{1},R_{2}]\times [R_{1},R_{2}]\) by an argument similar to the proof of [17, Lemmas 2.1 and 2.2], and therefore the linear problem
can be equivalently written as
Clearly, (H2) yields that \(T: C[R_{1},R_{2}]\to C[R_{1},R_{2}]\) is a completely continuous operator. By (2.1) and (2.2) we get
which is a resonant problem. As this point, (2.3) can be transformed into the equivalent integraldifferential equation
where the function χ is given as in (H1). In the following, we concentrate on the existence of positive solutions of (2.4). To this end, we denote by \(G(t,s)\) the Green’s function of the problem
Then by applying the same approach as in the proofs of [17, Lemmas 2.1 and 2.2] we can show that \(G(t,s)>0\) on \([R_{1},R_{2}]\times [R_{1},R_{2}]\) and (2.4) can be rewritten as the equivalent integral equation
Let E be the Banach space
equipped with the norm
Denote by \(m_{G}\) and \(M_{G}\) the minimum and maximum of \(G(t,s)\) on \([R_{1},R_{2}]\times [R_{1},R_{2}]\), respectively. Set \(\sigma =\frac{m_{G}}{M_{G}}\) and
Then \(0<\sigma <1\), and \(\mathcal{P}\) is a positive cone in E.
Lemma 2.1
Assume (H1) and (H2). Then \(A(\mathcal{P})\subseteq \mathcal{P}\), and \(A: \mathcal{P}\to \mathcal{P}\) is completely continuous.
Proof
Using (H1) and (H2), for any \(u\in \mathcal{P}\), we get
and therefore \(\Au\\leq M_{G}\cdot \int _{R_{1}}^{R_{2}} \{cp(s)u(s)T(u(s))+ (p(s)f(s,u(s))+\chi (s)u(s) ) \}\,ds\). On the other hand,
Combining the above two inequalities, we obtain \(Au(t)\geq \sigma \Au\\). Hence \(A(\mathcal{P})\subseteq \mathcal{P}\). Finally, using (H1)–(H2), in a standard way, we can easily show that \(A:\mathcal{P}\to \mathcal{P}\) is completely continuous. □
The main tool adopted in the paper is the following:
Lemma 2.2
([18])
Let E be a Banach space, and let \(\mathcal{P}\subseteq E\) be a cone. Let \(\Omega _{1}\) and \(\Omega _{2}\) be open bounded subsets of E satisfying \(0\in \Omega _{1}\) and \(\bar{\Omega }_{1}\subseteq \Omega _{2}\), and let \(T:\mathcal{P}\cap (\bar{\Omega }_{2}\setminus \Omega _{1})\to \mathcal{P}\) be a completely continuous operator such that

(i)
\(\Tu\\leq \u\\), \(u\in \mathcal{P}\cap \partial \Omega _{1}\), and \(\Tu\\geq \u\\), \(u\in \mathcal{P}\cap \partial \Omega _{2}\),
or

(ii)
\(\Tu\\geq \u\\), \(u\in \mathcal{P}\cap \partial \Omega _{1}\), and \(\Tu\\leq \u\\), \(u\in \mathcal{P}\cap \partial \Omega _{2}\).
Then T has a fixed point in \(\mathcal{P}\cap (\bar{\Omega }_{2}\setminus \Omega _{1})\).
We conclude this section by giving some notations to be used later. Set
and
where \(K(t,s)\) is as before. Define
Then it is not difficult to see that \(p_{0}>0\).
Proof of main results
Proof of Theorem 1.1
For positive constants \(r< R\), set
Then \(\Omega _{1}\) and \(\Omega _{2}\) are open bounded subsets of E with \(0\in \Omega _{1}\) and \(\bar{\Omega }_{1}\subseteq \Omega _{2}\).
By (1.5) there exists \(r_{1}>0\) such that for any \(0< u\leq r_{1}\),
where \(\epsilon >0\) is a constant small enough so that \(\epsilon lM_{G}\leq \frac{1}{2}\), and \(M_{G}\) is defined as in Sect. 2. Thus for \(u\in \mathcal{P}\) with \(\u\\leq r_{1}\),
From \(g_{0}=0\) it follows there exists a positive constant
such that \(g(u)\leq \varepsilon u\) for any \(0< u\leq r_{2}\), and therefore for \(u\in \mathcal{P}\) satisfying \(\u\\leq r_{2}\), simple estimation shows that
where ε is a sufficiently small positive constant such that \(\varepsilon cMM_{G}p^{2}_{0}\leq \frac{1}{2}\), and \(p_{0}\) is given by (2.6). Let \(r=\min \{r_{1}, r_{2}\}\). Then for \(u\in \mathcal{P}\) with \(\u\=r\), we get
which implies \(\Tu\\leq \u\\) for \(u\in \mathcal{P}\cap \partial \Omega _{1}\).
On the other hand, \(f_{\infty }=\infty \) yields that there exists \(\tilde{R}>0\) such that
where \(\eta >0\) is a constant large enough with \(\sigma lm_{G}(\eta +\underline{\chi })\geq 1\). Fixing \(R>\max \{r, \frac{\tilde{R}}{\sigma } \}\) and letting \(u\in \mathcal{P}\) with \(\u\=R\), we have
and therefore
Therefore we can deduce from (H2) that for \(u\in \mathcal{P}\) with \(\u\=R\),
which shows that \(\Tu\\geq \u\\) for \(u\in \mathcal{P}\cap \partial \Omega _{2}\).
By Lemma 2.2(i) A possesses a fixed point in \(\mathcal{P}\cap (\bar{\Omega }_{2}\setminus \Omega _{1})\), which is just a positive solution of (2.4). Accordingly, it follows from (2.2) that the original elliptic system (1.1) admits at least one positive radial solution. □
Proof of Theorem 1.2
To apply Lemma 2.2, we adopt the same strategy and notations as before. First, we show that for \(r>0\) sufficiently small,
Indeed, by \(f_{0}=\infty \) there exists \(\tilde{r}>0\) such that
where \(\beta >0\) is a constant large enough with \(\sigma lm_{G}(\beta +\underline{\chi })\geq 1\). Thus, for \(0< r\leq \tilde{r}\), if \(u\in \mathcal{P}\) and \(\u\=r\), then
which, together with (H2)′, implies
Hence (3.2) holds.
Next, we prove that for \(R>0\) large enough,
From (1.6) it follows that there exists \(\tilde{R}>0\) such that
for \(u\geq \tilde{R}\), where \(\epsilon >0\) satisfies \(\epsilon lM_{G}\leq \frac{1}{2}\). Let \(\tilde{R}_{1}>\max \{\tilde{r}, \frac{\tilde{R}}{\sigma }\}\). Then for \(u\in \mathcal{P}\) with \(\u\\geq \tilde{R}_{1}\), we get
and thus
On the other hand, (H2)′ implies that there exists \(\tilde{R}_{2}>0\) such that \(p(t)g(u)\leq \varepsilon \) for any \(u\geq \tilde{R}_{2}\). Therefore, for \(u\in \mathcal{P}\) with \(\u\\geq \tilde{R}_{2}\), we have
where \(\varepsilon >0\) is a constant satisfying \(\varepsilon clMM_{G}p_{0}\leq \frac{1}{2}\). Let \(R=\max \{\tilde{R}_{1}, \tilde{R}_{2}\}\). Then for \(u\in \mathcal{P}\) with \(\u\=R\), we easily verify that
which yields (3.3).
Consequently, Lemma 2.2(ii) ensures that A has a fixed point in \(\mathcal{P}\cap (\bar{\Omega }_{2}\setminus \Omega _{1})\), and thus system (1.1) admits a positive radial solution. □
Remark 3.1
To illustrate the results of Theorem 1.1, we choose
Let \(g(u)=u^{\alpha }\), \(u\in [0,\infty )\), and
where \(\alpha >1\) is a constant. Then it is not hard to verify that the assumptions in Theorem 1.1 are all satisfied. Therefore elliptic system (1.1) admits at least one positive radial solution.
Remark 3.2
To estimate (3.4), we assume (H2)′ in Theorem 1.2. Nevertheless, we believe that system (1.1) may admit positive radial solutions under (H2) and some suitable conditions on the nonlinearity g, which will be treated in the forthcoming paper. Clearly, Theorems 1.1 and 1.2 apply to models that cannot be dealt with by the results in the existing literature, and thus our main results are novel.
In the rest of the section, we consider the elliptic system
where Ω is the annulus introduced in Sect. 1. Note that the boundary condition in (3.5) means that the nuclear reactors exchange heat energy with the outside and neutron flux cannot cross the boundary of the container, which is the case closer to the reality. In this case the positive constant α is called the heat transfer coefficient. Obviously, system (3.5) corresponds to the nuclear reactor model
For radial solutions, elliptic system (3.5) is equivalent to
Applying Lemma 2.2, by an argument similar to that of Sects. 2 and 3 we can show that the results of Theorems 1.1 and 1.2 are still valid for elliptic system (3.5).
Availability of data and materials
Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.
References
 1.
Kastenberg, W.E., Chambré, P.L.: On the stability of nonlinear spacedependent reactor kinetics. Nucl. Sci. Eng. 31, 67–79 (1968)
 2.
Pao, C.V.: On nonlinear reaction–diffusion systems. J. Math. Anal. Appl. 87, 165–198 (1982)
 3.
Pao, C.V.: Bifurcation analysis on a nonlinear diffusion system in reactor dynamics. Appl. Anal. 9, 107–119 (1979)
 4.
Gu, Y., Wang, M.: Existence of positive stationary solutions and threshold results for a reaction–diffusion system. J. Differ. Equ. 130, 277–291 (1996)
 5.
Arioli, G.: Long term dynamics of a reaction–diffusion system. J. Differ. Equ. 235, 298–307 (2007)
 6.
LópezGómez, J.: The steady states of a noncooperative model of nuclear reactors. J. Differ. Equ. 246, 358–372 (2009)
 7.
Wang, F., An, Y.: Positive solutions for a secondorder differential system. J. Math. Anal. Appl. 373, 370–375 (2011)
 8.
Wang, F., An, Y.: On positive solutions for a second order differential system with indefinite weight. Appl. Math. Comput. 259, 753–761 (2015)
 9.
Wang, F., Wang, Y.: Existence of positive stationary solutions for a reaction–diffusion system. Bound. Value Probl. 2016(11), 1 (2016)
 10.
Li, Y., Li, F.: Nontrivial solutions to a class of systems of secondorder differential equations. J. Math. Anal. Appl. 388, 410–419 (2012)
 11.
Chen, R., Ma, R.: Positive solutions of the secondorder differential systems in reactor dynamics. Appl. Math. Comput. 219, 3882–3892 (2012)
 12.
Chen, R., Ma, R.: Global bifurcation of positive radial solutions for an elliptic system in reactor dynamics. Comput. Math. Appl. 65, 1119–1128 (2013)
 13.
Hai, D.D., Shivaji, R.: Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball. J. Differ. Equ. 266, 2232–2243 (2019)
 14.
Ma, R.: Existence of positive solutions for superlinear semipositone mpoint boundaryvalue problems. Proc. Edinb. Math. Soc. 46, 279–292 (2003)
 15.
Anton, I., LópezGómez, J.: Steadystates of a noncooperative model arising in nuclear engineering. Nonlinear Anal., Real World Appl. 14, 1340–1360 (2013)
 16.
Anton, I., LópezGómez, J.: Dynamics of a parabolic problem arising in nuclear engineering. Differ. Integral Equ. 27, 691–720 (2014)
 17.
Chen, R., Lu, Y.: Existence and multiplicity of positive solutions to nonlinear semipositone Neumann boundary value problem. Ann. Differ. Equ. 28, 137–145 (2012)
 18.
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Acknowledgements
Not applicable.
Funding
The first author is supported by the NSFC (No. 11761004), the NSF of Ningxia Hui Autonomous Region of China (No. 2020AAC03232), the Scientific Research Funds of North Minzu University (No. 2021XYZSX01),and the Construction Project of FirstClass Disciplines in Ningxia Higher Education (No. NXYLXK2017B09).
Author information
Affiliations
Contributions
RC carried out the analysis and proof the main results and was a major contributor in writing the manuscript. JL and GZ participated in checking the proofs. XK checked the English grammar and typing errors in the text. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Abbreviations
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, R., Liu, J., Zhang, G. et al. Positive radial solutions for a noncooperative resonant nuclear reactor model with signchanging nonlinearities. J Inequal Appl 2021, 74 (2021). https://doi.org/10.1186/s13660021026110
Received:
Accepted:
Published:
MSC
 34B15
Keywords
 Noncooperative models
 Radial solutions
 Resonance
 Existence
 Fixed point