Existence of positive solutions for singular four-point boundary value problem with a p -Laplacian

Chunmei Miao; Junfang Zhao; Weigao Ge

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 4, page 957-973
  • ISSN: 0011-4642

Abstract

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In this paper we deal with the four-point singular boundary value problem ( φ p ( u ' ( t ) ) ) ' + q ( t ) f ( t , u ( t ) , u ' ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ' ( 0 ) - α u ( ξ ) = 0 , u ' ( 1 ) + β u ( η ) = 0 , where φ p ( s ) = | s | p - 2 s , p > 1 , 0 < ξ < η < 1 , α , β > 0 , q ∈ C [ 0 , 1 ] , q ( t ) > 0 , t ∈ ( 0 , 1 ) , and f ∈ C ( [ 0 , 1 ] × ( 0 , + ∞ ) × ℝ , ( 0 , + ∞ ) ) may be singular at u = 0 . By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.

How to cite

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Miao, Chunmei, Zhao, Junfang, and Ge, Weigao. "Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian." Czechoslovak Mathematical Journal 59.4 (2009): 957-973. <http://eudml.org/doc/37969>.

@article{Miao2009,
abstract = {In this paper we deal with the four-point singular boundary value problem \[ \{\left\lbrace \begin\{array\}\{ll\} (\phi \_p(u^\{\prime \}(t)))^\{\prime \}+q(t)f(t,u(t),u^\{\prime \}(t))=0,& t\in (0,1),\\ u^\{\prime \}(0)-\alpha u(\xi )=0, \quad u^\{\prime \}(1)+\beta u(\eta )=0, \end\{array\}\right.\} \] where $\phi _p(s)=|s|^\{p-2\}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q(t)>0$, $t\in (0,1)$, and $f\in C([0,1]\times (0,+\infty )\times \mathbb \{R\},(0,+\infty ))$ may be singular at $u = 0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.},
author = {Miao, Chunmei, Zhao, Junfang, Ge, Weigao},
journal = {Czechoslovak Mathematical Journal},
keywords = {singular; four-point; positive solution; $p$-Laplacian; singular; four-point; positive solution; -Laplacian},
language = {eng},
number = {4},
pages = {957-973},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian},
url = {http://eudml.org/doc/37969},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Miao, Chunmei
AU - Zhao, Junfang
AU - Ge, Weigao
TI - Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 4
SP - 957
EP - 973
AB - In this paper we deal with the four-point singular boundary value problem \[ {\left\lbrace \begin{array}{ll} (\phi _p(u^{\prime }(t)))^{\prime }+q(t)f(t,u(t),u^{\prime }(t))=0,& t\in (0,1),\\ u^{\prime }(0)-\alpha u(\xi )=0, \quad u^{\prime }(1)+\beta u(\eta )=0, \end{array}\right.} \] where $\phi _p(s)=|s|^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q(t)>0$, $t\in (0,1)$, and $f\in C([0,1]\times (0,+\infty )\times \mathbb {R},(0,+\infty ))$ may be singular at $u = 0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.
LA - eng
KW - singular; four-point; positive solution; $p$-Laplacian; singular; four-point; positive solution; -Laplacian
UR - http://eudml.org/doc/37969
ER -

References

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