S -shaped component of nodal solutions for problem involving one-dimension mean curvature operator

Ruyun Ma; Zhiqian He; Xiaoxiao Su

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 321-333
  • ISSN: 0011-4642

Abstract

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Let E = { u ∈ C 1 [ 0 , 1 ] : u ( 0 ) = u ( 1 ) = 0 } . Let S k ν with ν = { + , - } denote the set of functions u ∈ E which have exactly k - 1 interior nodal zeros in (0, 1) and ν u be positive near 0 . We show the existence of S -shaped connected component of S k ν -solutions of the problem u ' 1 - u ' 2 ' + λ a ( x ) f ( u ) = 0 , x ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter, a ∈ C ( [ 0 , 1 ] , ( 0 , ∞ ) ) . We determine the intervals of parameter λ in which the above problem has one, two or three S k ν -solutions. The proofs of the main results are based upon the bifurcation technique.

How to cite

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Ma, Ruyun, He, Zhiqian, and Su, Xiaoxiao. "$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator." Czechoslovak Mathematical Journal 73.2 (2023): 321-333. <http://eudml.org/doc/299511>.

@article{Ma2023,
abstract = {Let $E=\lbrace u\in C^1[0,1] \colon u(0)=u(1)=0\rbrace $. Let $S_k^\nu $ with $\nu =\lbrace +, -\rbrace $ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^\nu $-solutions of the problem \[ \{\left\lbrace \begin\{array\}\{ll\} \biggl (\dfrac\{u^\{\prime \}\}\{\sqrt\{1-u^\{\prime 2\}\}\}\bigg )^\{\prime \}+\lambda a(x) f(u)=0, & x\in (0,1), \\ u(0)=u(1)=0, & \end\{array\}\right.\} \] where $\lambda >0$ is a parameter, $a\in C([0, 1], (0,\infty ))$. We determine the intervals of parameter $\lambda $ in which the above problem has one, two or three $S_k^\nu $-solutions. The proofs of the main results are based upon the bifurcation technique.},
author = {Ma, Ruyun, He, Zhiqian, Su, Xiaoxiao},
journal = {Czechoslovak Mathematical Journal},
keywords = {mean curvature operator; $S_k^\nu $-solution; bifurcation; Sturm-type comparison theorem},
language = {eng},
number = {2},
pages = {321-333},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator},
url = {http://eudml.org/doc/299511},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Ma, Ruyun
AU - He, Zhiqian
AU - Su, Xiaoxiao
TI - $S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 321
EP - 333
AB - Let $E=\lbrace u\in C^1[0,1] \colon u(0)=u(1)=0\rbrace $. Let $S_k^\nu $ with $\nu =\lbrace +, -\rbrace $ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^\nu $-solutions of the problem \[ {\left\lbrace \begin{array}{ll} \biggl (\dfrac{u^{\prime }}{\sqrt{1-u^{\prime 2}}}\bigg )^{\prime }+\lambda a(x) f(u)=0, & x\in (0,1), \\ u(0)=u(1)=0, & \end{array}\right.} \] where $\lambda >0$ is a parameter, $a\in C([0, 1], (0,\infty ))$. We determine the intervals of parameter $\lambda $ in which the above problem has one, two or three $S_k^\nu $-solutions. The proofs of the main results are based upon the bifurcation technique.
LA - eng
KW - mean curvature operator; $S_k^\nu $-solution; bifurcation; Sturm-type comparison theorem
UR - http://eudml.org/doc/299511
ER -

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