Multiple positive solutions for the one-dimensional \(p\)-Laplacian. (English) Zbl 0961.34012

Kong, Lingbin; Wang, Junyu

Nonlinear Anal., Theory Methods Appl. 42, No. 8, 1327-1333 (2000).

The existence of multiple positive solutions to the one-dimensional \(p\)-Laplacian \[ (( \Phi_{p} (u'))' + h(t) f(u) =0 \] subject to one of the following three pairs of nonlinear boundary conditions \[ u(0) -g_{1} (u'(0))=0, \quad u(1) +g_{2} (u'(1)) =0, \]
\[ u(0) - g_{1} (u'(0)) =0, \quad u(1)=0, \]
\[ u'(0) =0, \quad u(1) +g_{2} (u'(1)) =0, \] with \(\Phi_{p} (v) = |v|^{p-2} v\), \(p >1\), \(h(t)\) is a nonnegative measurable function on \((0,1)\), \(f(u) \) is a nonnegative continuous function on \([0, +\infty) \), and \(g_{1} (v) \) and \(g_{2} (v) \) are all continuous functions defined on \((- \infty, + \infty) \) , is studied. Results from papers of the reference are improved and generalized.

Reviewer: Georgiev Lyuben Vulkov (Rousse)

MSC:

34B18

Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

one-dimensional \(p\)-Laplacian; fixed-point index; multiplicity of positive solutions

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