Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems

Bai, Dingyong, and Chen, Yuming. "Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems." Applications of Mathematics 60.4 (2015): 343-353. <http://eudml.org/doc/271572>.

@article{Bai2015,
abstract = {We discuss the discrete $p$-Laplacian eigenvalue problem, \[ \{\left\lbrace \begin\{array\}\{ll\} \Delta (\phi \_p(\Delta u(k-1)))+\lambda a(k)g(u(k))=0,\quad k\in \lbrace 1,2, \ldots , T\rbrace ,\\ u(0)=u(T+1)=0, \end\{array\}\right.\} \] where $T>1$ is a given positive integer and $\phi _p(x):=|x|^\{p-2\}x$, $p > 1$. First, the existence of an unbounded continuum $\mathcal \{C\}$ of positive solutions emanating from $(\lambda , u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal \{C\}$ is a monotone continuous curve globally defined for all $\lambda >0$.},
author = {Bai, Dingyong, Chen, Yuming},
journal = {Applications of Mathematics},
keywords = {discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method; discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method},
language = {eng},
number = {4},
pages = {343-353},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems},
url = {http://eudml.org/doc/271572},
volume = {60},
year = {2015},
}

TY - JOUR
AU - Bai, Dingyong
AU - Chen, Yuming
TI - Global continuum of positive solutions for discrete $p$-Laplacian eigenvalue problems
JO - Applications of Mathematics
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 4
SP - 343
EP - 353
AB - We discuss the discrete $p$-Laplacian eigenvalue problem, \[ {\left\lbrace \begin{array}{ll} \Delta (\phi _p(\Delta u(k-1)))+\lambda a(k)g(u(k))=0,\quad k\in \lbrace 1,2, \ldots , T\rbrace ,\\ u(0)=u(T+1)=0, \end{array}\right.} \] where $T>1$ is a given positive integer and $\phi _p(x):=|x|^{p-2}x$, $p > 1$. First, the existence of an unbounded continuum $\mathcal {C}$ of positive solutions emanating from $(\lambda , u)=(0,0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $\lambda >0$ and all solutions are ordered. Thus the continuum $\mathcal {C}$ is a monotone continuous curve globally defined for all $\lambda >0$.
LA - eng
KW - discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method; discrete $p$-Laplacian eigenvalue problem; positive solution; continuum; Picone-type identity; lower and upper solutions method
UR - http://eudml.org/doc/271572
ER -