American Institute of Mathematical Sciences

March  2021, 41(3): 1483-1506. doi: 10.3934/dcds.2020326

Existence of nodal solutions for the sublinear Moore-Nehari differential equation

 Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

Received  May 2020 Revised  August 2020 Published  March 2021 Early access  September 2020

Fund Project: * This work was supported by JSPS KAKENHI Grant Number 20K03686

We study the existence of symmetric and asymmetric nodal solutions for the sublinear Moore-Nehari differential equation, $u''+h(x, \lambda)|u|^{p-1}u = 0$ in $(-1, 1)$ with $u(-1) = u(1) = 0$, where $0<p<1$, $h(x, \lambda) = 0$ for $|x|<\lambda$, $h(x, \lambda) = 1$ for $\lambda\leq |x|\leq 1$ and $\lambda\in (0, 1)$ is a parameter. We call a solution $u$ symmetric if it is even or odd. For an integer $n\geq 0$, we call a solution $u$ an $n$-nodal solution if it has exactly $n$ zeros in $(-1, 1)$. For each integer $n\geq 0$ and any $\lambda\in (0, 1)$, we prove that the equation has a unique $n$-nodal symmetric solution with $u'(-1)>0$. For integers $m, n \geq 0$, we call a solution $u$ an $(m, n)$-solution if it has exactly $m$ zeros in $(-1, 0)$ and exactly $n$ zeros in $(0, 1)$. We show the existence of an $(m, n)$-solution for each $m, n$ and prove that any $(m, m)$-solution is symmetric.

Citation: Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326
References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, (Universitext), Springer, New York, 2011 [2] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. doi: 10.1016/0362-546X(86)90011-8. [3] A. Gritsans and F. Sadyrbaev, Extension of the example by Moore-Nehari, Tatra Mt. Math. Publ., 63 (2015), 115–127. doi: 10.1515/tmmp-2015-0024. [4] P. Hartman, Ordinary Differential Equations, 2nd edition, Birkhäuser, Boston, (1982). [5] R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353–1362. doi: 10.1090/S0002-9939-2011-11172-9. [6] R. Kajikiya, Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987–2003. doi: 10.1016/j.jde.2011.08.032. [7] R. Kajikiya, Non-even positive solutions of the one dimensional $p$-Laplace Emden-Fowler equation, Applied Mathematics Letters, 25 (2012), 1891–1895. doi: 10.1016/j.aml.2012.02.057. [8] R. Kajikiya, Non-even positive solutions of the Emden-Fowler equations with sign-changing weights, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 631–642. doi: 10.1017/S0308210511001594. [9] R. Kajikiya, Symmetric and asymmetric nodal solutions for the Moore-Nehari differential equation, Submitted for publication. [10] R. Kajikiya, I. Sim and S. Tanaka, Symmetry-breaking bifurcation for the Moore-Nehari differential equation, Nonlinear Differential Equations and Applications, 25 (2018), article 54. doi: 10.1007/s00030-018-0545-3. [11] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonlinear Stud., 15 (2015), 253–288. doi: 10.1515/ans-2015-0201. [12] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP's, Topol. Methods Nonlinear Anal., 49 (2017), 359–376. doi: 10.12775/tmna.2016.087. [13] J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 923–946. doi: 10.3934/dcdsb.2017047. [14] J. López-Gómez and P. H. Rabinowitz, The structure of the set of $1$-node solutions of a class of degenerate BVP's, J. Differential Equations, 268 (2020), 4691–4732. doi: 10.1016/j.jde.2019.10.040. [15] R. A. Moore and Z. Nehari, Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30–52. doi: 10.1090/S0002-9947-1959-0111897-8. [16] Y. Naito and S. Tanaka, On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear Anal., 56 (2004), 919–935. doi: 10.1016/j.na.2003.10.020. [17] D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467–480. doi: 10.1142/S0219199702000725.

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References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, (Universitext), Springer, New York, 2011 [2] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. doi: 10.1016/0362-546X(86)90011-8. [3] A. Gritsans and F. Sadyrbaev, Extension of the example by Moore-Nehari, Tatra Mt. Math. Publ., 63 (2015), 115–127. doi: 10.1515/tmmp-2015-0024. [4] P. Hartman, Ordinary Differential Equations, 2nd edition, Birkhäuser, Boston, (1982). [5] R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353–1362. doi: 10.1090/S0002-9939-2011-11172-9. [6] R. Kajikiya, Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987–2003. doi: 10.1016/j.jde.2011.08.032. [7] R. Kajikiya, Non-even positive solutions of the one dimensional $p$-Laplace Emden-Fowler equation, Applied Mathematics Letters, 25 (2012), 1891–1895. doi: 10.1016/j.aml.2012.02.057. [8] R. Kajikiya, Non-even positive solutions of the Emden-Fowler equations with sign-changing weights, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 631–642. doi: 10.1017/S0308210511001594. [9] R. Kajikiya, Symmetric and asymmetric nodal solutions for the Moore-Nehari differential equation, Submitted for publication. [10] R. Kajikiya, I. Sim and S. Tanaka, Symmetry-breaking bifurcation for the Moore-Nehari differential equation, Nonlinear Differential Equations and Applications, 25 (2018), article 54. doi: 10.1007/s00030-018-0545-3. [11] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonlinear Stud., 15 (2015), 253–288. doi: 10.1515/ans-2015-0201. [12] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP's, Topol. Methods Nonlinear Anal., 49 (2017), 359–376. doi: 10.12775/tmna.2016.087. [13] J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 923–946. doi: 10.3934/dcdsb.2017047. [14] J. López-Gómez and P. H. Rabinowitz, The structure of the set of $1$-node solutions of a class of degenerate BVP's, J. Differential Equations, 268 (2020), 4691–4732. doi: 10.1016/j.jde.2019.10.040. [15] R. A. Moore and Z. Nehari, Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30–52. doi: 10.1090/S0002-9947-1959-0111897-8. [16] Y. Naito and S. Tanaka, On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations, Nonlinear Anal., 56 (2004), 919–935. doi: 10.1016/j.na.2003.10.020. [17] D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467–480. doi: 10.1142/S0219199702000725.
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