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

Ryuji Kajikiya

doi:10.3934/dcds.2020326

Article Contents

2021, Volume 41, Issue 3: 1483-1506. Doi: 10.3934/dcds.2020326

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

Ryuji Kajikiya^,

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

Received: April 30, 2020

Revised: July 31, 2020

Early access: September 2020

Published: March 2021

^* This work was supported by JSPS KAKENHI Grant Number 20K03686

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Abstract

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.

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Mathematics Subject Classification: Primary: 34B15; Secondary: 34B08.

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