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Twin positive solutions for the one-dimensional \(p\)-Laplacian boundary value problems. (English) Zbl 1061.34013

The authors deal with the one-dimensional \(p\)-Laplacian \[ (g(u'))'+ e(t) f(u)= 0 \] subject to one of the following three pairs of nonlinear boundary conditions:
a) \(u(0)= B_0(u'(0))= 0\), \(u(1)+ B_1(u'(1))= 0\),
b) \(u(0)= B_0(u'(0))= 0\), \(u'(1)= 0\),
c) \(u'(0)= 0\), \(u(1)+ B_1(u'(1))= 0\), where \(g(v):= |v|^{p-2} v\), \(p> 1\).
The authors show the existence of at least two positive solutions. To this end, they use a fixed-point theorem in cones.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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