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Zeitschrift für Analysis und ihre Anwendungen


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Volume 26, Issue 1, 2007, pp. 25–41
DOI: 10.4171/ZAA/1308

Published online: 2007-03-31

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

Haishen Lü and Yi Xie

Positive solutions are obtained for the boundary value problem \begin{alignat*}{2} \left\{ \begin{aligned} -\Delta _{p}u&= \lambda (u^{\beta }+\tfrac{1}{u^{\alpha }}) &\; &\hbox{in } \Omega \\ u&> 0 &\; &\hbox{in } \Omega \\ u&= 0 &\; &\hbox{on }\partial \Omega\,, \end{aligned}% \right. \tag{$*$} \end{alignat*} where $\Delta _{p}u=\dive(| \nabla u| ^{p-2}\nabla u)$, $10$ is a real parameter. We obtain that Problem ($*$) has two positive weakly solutions if $\lambda $ is small enough.

Keywords: p-Laplacian, positive solution, critical point theory

Lü Haishen, Xie Yi: Existence and Multiplicity of Positive Solutions for Singular p-Laplacian Equations. Z. Anal. Anwend. 26 (2007), 25-41. doi: 10.4171/ZAA/1308