Geodesic
Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 4, pp. 957-973 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper we deal with the four-point singular boundary value problem $$ \begin {cases} (\phi _p(u'(t)))'+q(t)f(t,u(t),u'(t))=0, t\in (0,1),\\ u'(0)-\alpha u(\xi )=0, \quad u'(1)+\beta u(\eta )=0, \end {cases} $$ 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.
In this paper we deal with the four-point singular boundary value problem $$ \begin {cases} (\phi _p(u'(t)))'+q(t)f(t,u(t),u'(t))=0, t\in (0,1),\\ u'(0)-\alpha u(\xi )=0, \quad u'(1)+\beta u(\eta )=0, \end {cases} $$ 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.
Classification : 34B10, 34B16, 34B18
Keywords: singular; four-point; positive solution; $p$-Laplacian 
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     author = {Miao, Chunmei and Zhao, Junfang and Ge, Weigao},
     title = {Existence of positive solutions for singular four-point boundary value problem with a $p${-Laplacian}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {957--973},
     year = {2009},
     volume = {59},
     number = {4},
     mrnumber = {2563569},
     zbl = {1224.34053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2009_59_4_a6/}
}
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Miao, Chunmei; Zhao, Junfang; Ge, Weigao. Existence of positive solutions for singular four-point boundary value problem with a $p$-Laplacian. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 4, pp. 957-973. http://geodesic.mathdoc.fr/item/CMJ_2009_59_4_a6/