JournalszaaVol. 26, No. 1pp. 25–41

Existence and Multiplicity of Positive Solutions for Singular <em>p</em>-Laplacian Equations

  • Haishen Lü

    Hohai University, Nanjing, China
  • Yi Xie

    Hohai University, Nanjing, China
Existence and Multiplicity of Positive Solutions for Singular <em>p</em>-Laplacian Equations cover
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Abstract

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 Δpu=\dive(up2u)\Delta _{p}u=\dive(| \nabla u| ^{p-2}\nabla u), 1<p<N1<p<N, N3N\geq 3, ΩRN\Omega \subset {\mathbb R}^{N} is a bounded domain, 0<α<10<\alpha <1 and p1<β<p1p-1<\beta <p^{\ast }-1 \big(p=NpNpp^{\ast }=\frac{Np}{N-p}\big) are two constants, λ>0\lambda >0 is a real parameter. We obtain that Problem (*) has two positive weakly solutions if λ\lambda is small enough.

Cite this article

Haishen Lü, Yi Xie, Existence and Multiplicity of Positive Solutions for Singular <em>p</em>-Laplacian Equations. Z. Anal. Anwend. 26 (2007), no. 1, pp. 25–41

DOI 10.4171/ZAA/1308