Abstract
We study the one-dimensional nonlocal elliptic equation
where \(A = A(y)\) and \(B = B(y)\) are continuous functions, satisfying \(A(y) > 0\), \(B(y) > 0\) for \(y > 0\), \(p \ge 1\), \(q \ge 1\), and \(r > 1\) are given constants, and \(\lambda > 0\) is a bifurcation parameter. We establish the global behavior of solution curves and precise asymptotic formulas for \(u_{\lambda}(x)\) as \(\lambda \to \infty \).
Similar content being viewed by others
1 Introduction
We consider the following one-dimensional nonlocal elliptic equation
where \(A = A(y)\) and \(B = B(y)\) are continuous functions with \(A(y) > 0\), \(B(y) > 0\) for \(y > 0\), while \(p \ge 1\), \(q \ge 1\), \(r > 1\) are given constants, \(\Vert \cdot \Vert _{m}\) (\(m \ge 1\)) denotes the usual \(L^{m}\)-norm of the real-valued functions on I, and \(\lambda > 0\) is a bifurcation parameter. In this paper, we consider the following typical three cases.
-
(i)
\(A(y) = y\), \(B(y) = y\),
-
(ii)
\(A(y) = e^{y}\), \(B(y) = 1\), \(p = 2\),
-
(iii)
\(A(y) = e^{y}\), \(B(y) = y\).
The case (i) is a modification of a nonlocal problem of Kirchhoff type, which is motivated by the following problem (1.2) in [13]
Besides, the cases (ii) and (iii) are motivated by the mean field equation and nonlocal Liouville-type equations.
Nonlocal problems have been of interest to many researchers from mathematical point of view, since many problems are derived from the phenomena of relevant physical, biological, and engineering problems. Therefore, nonlocal problems have been widely studied by many authors. We refer the reader to Goodrich [7–9], Lacey [11, 12], Stańczy [14], [2–4, 10], and the references therein. One of the main interest in this area is the existence of positive solutions. On the other hand, there seems to be a few studies on bifurcation problems. We refer the reader to [17] and the references therein. Roughly speaking, in [17], the case \(A(y) = y^{j} + b\) and \(B(y) = y^{k}\), where \(j \ge 0\), \(k > 0\) are constants, has been considered, and the existence of a branch of positive solutions bifurcating from infinity at \(\lambda = 0\) has been discussed. Recently, the case, where \(A(y) = y + b\), \(B(y) \equiv 1\) and \(p = 2\), has been studied in [15], where \(b > 0\) is a constant, and the precise global behavior of solution curve has been obtained. For a standard bifurcation problems, we refer to [5].
The purpose of this paper is to consider more general nonlocal terms motivated by equations having background in physics and obtain the precise asymptotic behavior of bifurcation curves \(\lambda = \lambda (\alpha )\) and \(u_{\lambda}\) as \(\lambda \to \infty \). Here, \(\alpha := \alpha _{\lambda }=\Vert u_{\lambda}\Vert _{\infty}\) for given \(\lambda > 0\). The main tool here is time map method, also known as quadrature technique (cf. [12]). One can see the simple example of time map method in the Appendix.
Now, we state our main Theorems 1.1, 1.2, and 1.3. To do this, we prepare the following notation. For \(r > 1\), let
We know from [6] that there exists a unique solution \(W_{r}(x)\) of (1.3). For \(m \ge 1\), we put
We note that \(L_{m}\) is finite since \(\sqrt{1 - s^{2}} \le \sqrt{1 - s^{m+1}}\) for \(0 \le s \le 1\). Then, we have
Equation (1.6) has been obtained in [16]. For completeness, the proof of (1.5) and (1.6) will be given in the Appendix.
We begin with the first main Theorem 1.1, which will be proved in Sect. 2.
Theorem 1.1
Let \(A(y) = y\), \(B(y) = y\) in (1.1). Assume that \(p-q-r+1 \neq 0\). Then, there exists a unique solution \(u_{\lambda}(x)\) of (1.1) for any \(\lambda > 0\), and it is represented as
Further, λ is represented as the function of \(\alpha := \Vert u_{\lambda}\Vert _{\infty}\) as
Next, we consider the case \(A(y) = e^{y}\), \(B(y) = 1\) and \(p = 2\) and state Theorem 1.2. The proof will be given in Sect. 3. For \(r > 1\), we put
Theorem 1.2
Let \(A(y) = e^{y}\), \(B(y) = 1\) and \(p = 2\). For \(r > 1\), put \(\lambda _{r}:= e^{R_{r}}\).
-
(i)
If \(0 < \lambda < \lambda _{r}\), then there are no solutions of (1.1).
-
(ii)
If \(\lambda = \lambda _{r}\), then (1.1) has a unique solution \(u_{1,\lambda}\).
-
(iii)
If \(\lambda > \lambda _{r}\), then there are exactly two solutions \(u_{1,\lambda}\), \(u_{2,\lambda}\) with \(u_{1,\lambda}(x) < u_{2,\lambda}(x)\) for \(x \in I\).
-
(iv)
Let \(\lambda > \lambda _{r}\) be fixed. Then, there are two numbers \(\alpha _{1,\lambda}:= \Vert u_{1,\lambda}\Vert _{\infty}\) and \(\alpha _{2,\lambda}:= \Vert u_{2,\lambda}\Vert _{\infty}\) satisfying:
$$ \lambda = 2(r+1)L_{r}^{2} A\bigl(4M_{r,2}L_{r} \alpha _{j,\lambda}^{2}\bigr) \alpha _{j,\lambda}^{1-r}\quad (j = 1,2). $$(1.10)If \(\lambda = \lambda _{r}\), then \(\alpha _{0}:=\alpha _{1,\lambda} = \alpha _{2,\lambda}\) in (1.10).
-
(v)
As \(\lambda \to \infty \),
$$\begin{aligned}& u_{1,\lambda}(x) = \lambda ^{-1/(r-1)} \biggl\{ 1 + \frac{1}{r-1} \bigl\Vert W_{r}' \bigr\Vert _{2}^{2} \lambda ^{-2/(r-1)}\bigl(1 + o(1)\bigr) \biggr\} \bigl\Vert W_{r}' \bigr\Vert _{2}^{-1}W_{r}(x), \end{aligned}$$(1.11)$$\begin{aligned}& u_{2,\lambda}(x) = \biggl\{ \log \lambda + \frac{r-1}{2}\log (\log \lambda ) \bigl(1 + o(1)\bigr) \biggr\} ^{1/2} \bigl\Vert W_{r}' \bigr\Vert _{2}^{-1}W_{r}(x). \end{aligned}$$(1.12)
We finally state Theorem 1.3, which will be proved in Sect. 4.
Theorem 1.3
Let \(A(y) = e^{y}\), \(B(y) = y\). Let
and \(\lambda _{0}:= e^{C_{0}}\).
-
(i)
If \(0 < \lambda < \lambda _{0}\), then there are no solutions of (1.1).
-
(ii)
If \(\lambda = \lambda _{0}\), then (1.1) has a unique solution \(u_{1,\lambda}\).
-
(iii)
If \(\lambda > \lambda _{0}\), then there are exactly two solutions \(u_{1,\lambda}\), \(u_{2,\lambda}\) with \(u_{1,\lambda}(x) < u_{2,\lambda}(x)\) for \(x \in I\).
-
(iv)
As \(\lambda \to \infty \)
$$\begin{aligned}& u_{1,\lambda}(x) = \lambda ^{-1/(q+r-1))} \bigl\Vert W_{r}' \bigr\Vert _{q}^{-q/(q+r-1)} \\ \end{aligned}$$(1.14)$$\begin{aligned}& \hphantom{u_{1,\lambda}(x) ={}}{} \times \biggl\{ 1 + \frac{1}{q+r-1} \bigl\Vert W_{\lambda}' \bigr\Vert _{p}^{p} \bigl\Vert W_{\lambda}' \bigr\Vert _{q}^{pq/(q+r-1)} \lambda ^{-p/(q+r-1)}\bigl(1 + o(1)\bigr) \biggr\} W_{r}(x), \\& \alpha _{1,\lambda} = \lambda ^{-1/(q+r-1))} \bigl\Vert W_{r}' \bigr\Vert _{q}^{-q/(q+r-1)} \end{aligned}$$(1.15)$$\begin{aligned}& \hphantom{\alpha _{1,\lambda} ={}}{} \times \biggl\{ 1 + \frac{1}{q+r-1} \bigl\Vert W_{\lambda}' \bigr\Vert _{p}^{p} \bigl\Vert W_{\lambda}' \bigr\Vert _{q}^{pq/(q+r-1)} \lambda ^{-p/(q+r-1)}\bigl(1 + o(1)\bigr) \biggr\} \Vert W_{\lambda} \Vert _{\infty}, \end{aligned}$$(1.16)$$\begin{aligned}& u_{2,\lambda}(x) = \bigl\Vert W_{r}' \bigr\Vert _{q}^{-1}(\log \lambda )^{1/p} \biggl\{ 1 + \frac{p^{2}}{(q+r-1)^{3}} \frac{\log (\log \lambda )}{\log \lambda}\bigl(1 + o(1)\bigr) \biggr\} W_{r}(x), \\& \alpha _{2,\lambda} = \bigl\Vert W_{r}' \bigr\Vert _{q}^{-1}(\log \lambda )^{1/p} \biggl\{ 1 + \frac{p^{2}}{(q+r-1)^{3}} \frac{\log (\log \lambda )}{\log \lambda}\bigl(1 + o(1)\bigr) \biggr\} \Vert W_{ \lambda} \Vert _{\infty}. \end{aligned}$$(1.17)
The rest of this paper is organized as follows. We prove Theorems 1.1, 1.2, and 1.3 in Sects. 2, 3, and 4, respectively. In the proofs, the time map method and the argument from [1] play important roles. Finally, we prove (1.5) and (1.6) in the Appendix.
2 Proof of Theorem 1.1
Let \(\lambda > 0\) be fixed. We write \(w_{r}(x)\) as a unique solution of
We look for the solution \(u_{\lambda}(x)\) of the form
where \(t_{\lambda }> 0\) is a constant. By (1.1) and (2.1), we have
Since \(w_{r}(x) = \lambda ^{-1/(r-1)}W_{r}(x)\), we find from (2.3) that if
then (2.2) satisfies (1.1). By this, we have
By this and (2.2), we have
This implies (1.7). Since \(\alpha = \Vert u_{\lambda}\Vert _{\infty }= u_{\lambda}(1/2)\), we put \(x = 1/2\) in (2.6). Then, we obtain
By this, we obtain (1.8). Thus, the proof of Theorem 1.1 is complete.
3 Proof of Theorem 1.2
We first show the existence of \(u_{\lambda}\). We follow the argument in [1]. Let \(t > 0\) and \(A(t) = e^{t}\). We consider the following equation for \(t > 0\).
Assume that \(t_{\lambda }> 0\) satisfies (3.1). We put \(\gamma := t_{\lambda}^{1/2}\Vert w_{r}'\Vert _{2}^{-1}\) and \(u_{\lambda}:= \gamma w_{r}\). Then, we have
By this, we have
Since \(w_{r} = \lambda ^{1/(1-r)}W_{r}\), we have
On the other hand, suppose that \(u_{\lambda}\) satisfies (3.2). Then by putting \(t_{\lambda }= \Vert u_{\lambda}'\Vert _{2}^{2}\), we see that \(t_{\lambda}\) satisfies (3.1). Therefore, the number of the positive solutions t of the equation
coincide with the number of the solutions of (3.2). So, we solve the equation (3.1). To do this, for \(r > 1\), we set
Lemma 3.1
For \(r > 1\), let \(\lambda _{r}:= e^{R_{r}}\).
-
(i)
If \(0 < \lambda < \lambda _{r}\), then (3.4) has no solution.
-
(ii)
If \(\lambda = \lambda _{r}\), then (3.4) has a unique solution \(t_{0}\).
-
(iii)
If \(\lambda > \lambda _{r}\), then (3.4) has exactly two solutions \(t_{\lambda ,1}\), \(t_{\lambda ,2}\) with \(0 < t_{\lambda ,1} < t_{0} < t_{\lambda ,2}\).
Proof
We put \(K_{\lambda ,r}:= \lambda \Vert W_{r}'\Vert _{2}^{1-r}\). By (3.4), we have the equation
Since \(g'(t) = \frac{r-1}{2t}\), we see that \(g'(t_{0}) = 1\), where \(t_{0} = \frac{r-1}{2}\). Then the tangent line of \(y = g(t)\) at \((t_{0}, g(t_{0}))\) is \(g(t) - g(t_{0}) = t - t_{0}\). We see from this that if \(g(t_{0}) = t_{0}\), namely,
then the tangent line of \(g(t)\) at \(t = t_{0}\) is exactly the line \(y = t\). Equation (3.7) implies that
namely, \(\lambda = e^{R_{r}}\). This implies (ii). Since logt is a concave function w.r.t. \(t > 0\), and \(\log K_{\lambda , r}\) is increasing function of \(\lambda > 0\), if \(0 < \lambda < e^{R_{r}}\) (resp. \(\lambda > e^{R_{r}}\)), then we obtain (i) and (iii), respectively. Thus, the proof is complete. □
Proof of Theorem 1.2
By Lemma 3.1, we obtain Theorem 1.2(i), (ii), and (iii). Now, we show (iv). Assume that \(u_{\lambda}(x)\) is a solution of (1.1) for some \(\lambda > 0\). We write \(A = e^{\Vert u_{\lambda}'\Vert _{2}^{2}}\). By (1.1), we have
Recall that \(\alpha := \Vert u_{\lambda}\Vert _{\infty}\). Then (3.9) implies that
We know that \(u_{\lambda}(x)\) is a positive solution of \(-u''_{\lambda}(x) = (\lambda /A)u_{\lambda}(x)^{r}\) with the condition \(u_{\lambda}(0) = u_{\lambda}(1) = 0\). Therefore, by the result of Gidas, Ni, and Nirenberg [6], we know that \(u_{\lambda}(x) = u_{\lambda}(1-x)\) (\(0 \le x \le 1/2\)). By this, (3.10) implies that for \(0 \le x \le 1/2\),
By this, we have
By this, we have
By (3.11), we have
By this, we have
By this and (3.15), we have
This implies that
Thus, the proof of (iv) is complete. □
Now, we prove Theorem 1.2(v).
Lemma 3.2
As \(\lambda \to \infty \),
Proof
We first prove (3.19). Since \(t_{0} = (r-1)/2\), by (3.6), we see that \(t_{\lambda ,1} \to 0\) and \(t_{\lambda ,2} \to \infty \) as \(\lambda \to \infty \). By (3.4), we have
This implies that
where \(\delta \to 0\) as \(\lambda \to \infty \). By this and (3.6), we have
By this and the Taylor expansion, we have
By this, we have
This implies that as \(\lambda \to \infty \)
By this and (3.3), we obtain (3.19). Now, we prove (3.20). Since \(t_{\lambda ,2} \to \infty \) as \(\lambda \to \infty \), we have
This implies that
where \(\epsilon \to 0\) as \(\lambda \to \infty \). By this and (3.21), we obtain
By this, we obtain
By this, (3.3) and (3.28), we obtain (3.20). Thus, the proof is complete. □
4 Proof of Theorem 1.3
Let \(\lambda > 0\) be fixed. In what follows, C denotes various constants independent of λ. Following the idea of (3.1)–(3.3), we look for the solution of the form
If (4.1) is the solution of (1.1) with \(A(\Vert u_{\lambda}'\Vert _{p}^{p}) = e^{\Vert u_{\lambda}'\Vert _{p}^{p}}\) and \(B(\Vert u_{\lambda}'\Vert _{q}^{q}) = \Vert u_{\lambda}'\Vert _{q}^{q}\), then we have
This implies that
We put \(s:= t^{q+r-1}\). By taking log of the both side of (4.3), we have
where
We put
We look for \(s > 0\) satisfying \(g(s) = 0\). To do this, we consider the graph of \(g(s)\). We know that
By this, we find that \(g'(s_{0}) = 0\), where
By an elementary calculation, we see that if \(0 < s < s_{0}\) (resp. \(s > s_{0}\)), then \(g(s)\) is strictly decreasing (resp. strictly increasing) and \(g(s_{0})\) is the minimum value of \(g(s)\). By (4.5), (4.6), (4.8), and direct calculation, we have
We put \(\lambda _{1}:= e^{C_{0}}\). Then, \(g(s_{0}) > 0\) if \(0 < \lambda < \lambda _{1}\), \(g(s_{0}) = 0\) if \(\lambda = \lambda _{1}\) and \(g(s_{0}) < 0\) if \(\lambda > \lambda _{1}\). Then, we see that if \(0 < \lambda < \lambda _{1}\), then (4.6) (namely, (4.3) and (4.4)) has no solution, and if \(\lambda = \lambda _{1}\), then (4.6) (namely, (4.3) and (4.4)) has a unique solution \(s_{0}\), and if \(\lambda > \lambda _{1}\), then (4.6) (namely, (4.3) and (4.4)) has exactly two solutions \(s_{1}\), \(s_{2}\) with \(s_{1} < s_{0} < s_{2}\).
We see from the argument above that Theorem 1.3(i), (ii) hold. Moreover, let \(t_{\lambda ,j}:= s_{j}^{1/(q+r-1)}\) (\(j = 1,2\)). By this and (4.1), we obtain Theorem 1.3(iii).
Now, we consider the case (iv). Since it is difficult to obtain \(t_{\lambda ,j}:= s_{j}^{1/(q+r-1)}\) (\(j = 1,2\)) exactly, we first establish the asymptotic formula for \(t_{\lambda ,j}\) for \(\lambda \gg 1\).
Lemma 4.1
Assume that \(\lambda \gg 1\). Then,
Proof
We put \(s_{\lambda ,2}:= t_{\lambda ,2}^{q+r-1}\). By (4.3), we have
By this, we have
Then, three cases should be considered.
Case 1. Assume that there exists a subsequence of \(\{\lambda \}\), which is denoted by \(\{\lambda \}\) again, such that as \(\lambda \to \infty \),
Then, by this and (4.12), we have
This implies that
By this and (4.8), we have \(s_{0} > s_{\lambda ,2}\). This is a contradiction.
Case 2. Assume that there exists a subsequence of \(\{\lambda \}\), which is denoted by \(\{\lambda \}\) again, such that as \(\lambda \to \infty \),
Then, by (4.12), we have
This implies that
By this, (4.12), and (4.18), we have
This is a contradiction.
Case 3. Therefore, there exists a subsequence of \(\{\lambda \}\), which is denoted by \(\{\lambda \}\) again, such that as \(\lambda \to \infty \),
By this and taking a subsequence of \(\{\lambda \}\) again if necessary, we see that there exists a constant \(C_{4} > 0\) such that as \(\lambda \to \infty \),
where \(\delta _{1} \to 0\) as \(\lambda \to \infty \). By this and (4.12), we have
This implies that
where \(\delta _{1} \to 0\) as \(\lambda \to \infty \). This implies that
Namely, \(C_{4} = \Vert W_{r}'\Vert _{p}^{-(q+r-1)}\). By (4.22) and (4.23), we have
By this, we have
By this, (4.23), and the Taylor expansion, we have
Indeed, we see from (4.27) that \(s_{0} < s_{\lambda ,2}\). Therefore, by (4.27), we obtain (4.10). Thus, the proof is complete. □
Lemma 4.2
Assume that \(\lambda \gg 1\). Then
Proof
Since \(s_{\lambda ,1} < s_{0}\), we find from Lemma 4.1 that as \(\lambda \to \infty \),
By this and (4.12), we have
This implies that
By this and (4.8), we see that \(s = s_{\lambda ,1}\) is determined by (4.31). By (4.31), we have
as \(\lambda \to \infty \). Now, we calculate \(s_{1}\). By (4.30) and (4.12), we have
By this, for \(\lambda \gg 1\), we have
where \(\eta \to 0\) as \(\lambda \to \infty \). By this, (4.12), and the Taylor expansion, we have
By this and (4.34), we have
By this, (4.20), and the Taylor expansion, we have
By this, we have
Thus, we obtain (4.28). □
Proof of Theorem 1.3
By Lemma 4.2, for \(\lambda \gg 1\), we obtain
This implies (1.14). Further, by Lemma 4.1, we obtain
This implies (1.16). To obtain (1.15) and (1.17), we just put \(x = 1/2\) in (4.39) and (4.40). Thus, the proof is complete. □
Availability of data and materials
Not applicable.
References
Alves, C.O., Corréa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Arcoya, D., Leonori, T., Primo, A.: Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem. Acta Appl. Math. 127, 87–104 (2013)
Corrêa, F.J.S.A.: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59, 1147–1155 (2004)
Corrêa, F.J.S.A., de Morais Filho, D.C.: On a class of nonlocal elliptic problems via Galerkin method. J. Math. Anal. Appl. 310(1), 177–187 (2005)
Fraile, J.M., López-Gómez, J., Sabina de Lis, J.: On the global structure of the set of positive solutions of some semilinear elliptic boundary value problems. J. Differ. Equ. 123, 180–212 (1995)
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979)
Goodrich, C.S.: A topological approach to nonlocal elliptic partial differential equations on an annulus. Math. Nachr. 294, 286–309 (2021)
Goodrich, C.S.: Differential equations with multiple sign changing convolution coefficients. Int. J. Math. 32(8), Paper No. 2150057, 28 pp. (2021)
Goodrich, C.S.: An analysis of nonlocal difference equations with finite convolution coefficients. J. Fixed Point Theory Appl. 24(1), Paper No. 1, 19 pp. (2022)
Lacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. I. Model derivation and some special cases. Eur. J. Appl. Math. 6, 127–144 (1995)
Lacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. II. General proof of blow-up and asymptotics of runaway. Eur. J. Appl. Math. 6, 201–224 (1995)
Laetsch, T.: The number of solutions of a nonlinear two point boundary value problem. Indiana Univ. Math. J. 20, 1–13 (970/1971)
Liang, Z., Li, F., Shi, J.: Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31(1), 155–167 (2014)
Shibata, T.: Global and asymptotic behaviors of bifurcation curves of one-dimensional nonlocal elliptic equations. J. Math. Anal. Appl. 516, 126525 (2022)
Shibata, T.: Bifurcation diagrams of one-dimensional Kirchhoff type equations. Adv. Nonlinear Anal. (to appear)
Stańczy, R.: Nonlocal elliptic equations. Nonlinear Anal. 47, 3579–3584 (2001)
Wang, W., Tang, W.: Bifurcation of positive solutions for a nonlocal problem. Mediterr. J. Math. 13, 3955–3964 (2016)
Acknowledgements
Not applicable.
Authors’ information
Not applicable.
Funding
This work was supported by JSPS KAKENHI Grant Number JP21K03310.
Author information
Authors and Affiliations
Contributions
The author proved all the theorems. He also read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare no competing interests.
Appendix
Appendix
Let \(r > 1\). We first show (1.6), which was proved in [14] for completeness. We apply the time map argument to (1.3), cf. [12]. Since (1.3) is autonomous, we have
By (1.3), for \(0 \le x \le 1\), we have
By this and (A.3), we have
By this and (A.2), for \(0 \le x \le 1/2\), using \(\theta = \xi s\), we have
By this, we have
This implies (1.6). We next show (1.5). By (A.1), (A.3), and (A.6), we have
This implies (1.5).
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
Shibata, T. Asymptotic behavior of solution curves of nonlocal one-dimensional elliptic equations. Bound Value Probl 2022, 63 (2022). https://doi.org/10.1186/s13661-022-01644-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-022-01644-8