    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  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 & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020326
##### References:
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##### References:
  H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, (Universitext), Springer, New York, 2011 Google Scholar  H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. doi: 10.1016/0362-546X(86)90011-8.  Google Scholar  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.  Google Scholar  P. Hartman, Ordinary Differential Equations, 2nd edition, Birkhäuser, Boston, (1982). Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Kajikiya, Symmetric and asymmetric nodal solutions for the Moore-Nehari differential equation, Submitted for publication. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar
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