Existence and nonexistence of positive eigenfunctions for the 𝑝-Laplacian

Binding, P. A.; Huang, Y. X.

doi:10.1090/S0002-9939-1995-1260160-8

Existence and nonexistence of positive eigenfunctions for the $p$-Laplacian
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by P. A. Binding and Y. X. Huang PDF

Proc. Amer. Math. Soc. 123 (1995), 1833-1838 Request permission

Abstract:

We study the relation between (i) the principal eigencurve, i.e., the graph of ${\mu _1}$ satisfying \begin{equation}\tag {$ \ast $} - {\operatorname {div}}(|\nabla u{|^{p - 2}}\nabla u) + (q(x) - \lambda w(x))|u{|^{p - 2}}u = {\mu _1}(\lambda )|u{|^{p - 2}}\mu \end{equation} on a smooth bounded domain $\Omega$ in ${{\mathbf {R}}^N}$ with $p > 1$, and (ii) existence and nonexistence of positive solutions of $( \ast )$ with ${\mu _1}(\lambda ) = 0$. Eigencurve arguments are used extensively.

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