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# Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations

Published:2006-06-01
Printed: Jun 2006
• Ravi P. Agarwal
• Daomin Cao
• Haishen Lü
• Donal O'Regan
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## Abstract

Positive solutions are obtained for the boundary value problem $\begin{cases} -( | u'| ^{p-2}u')' =\lambda f( t,u),\;t\in ( 0,1) ,p>1\\ u( 0) =u(1) =0. \end{cases}$ Here $f(t,u) \geq -M,$ ($M$ is a positive constant) for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper-lower solution method.
 Keywords: one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution
 MSC Classifications: 34B15 - Nonlinear boundary value problems