Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional \(p\)-Laplacian. (English) Zbl 1057.34012

Sánchez, Justino

J. Math. Anal. Appl. 292, No. 2, 401-414 (2004).

The author studies existence and multiplicity of positive solutions for the singular Dirichlet eigenvalue-type problem \[ -(\varphi_p(u'(t)))'= \lambda h(t) f(u(t)) \quad \text{in }(0,1), \qquad u(0)= u(1)= 0, \] where \(\varphi_p(\xi)=| \xi|^{p-2}\xi\), \(p>1\), and \(\lambda\) is a positive parameter, \(h(t)\) is a nonnegative measurable function on \((0,1)\) that may be singular at \(t=0\) and/or \(t=1\), \(f(u)\) is a nonnegative continuous function on \([0,+\infty)\), moreover \(f\) is either sublinear or superlinear at zero and/or \(+\infty\). To this end, the author uses the fixed-point index theory as well as a fixed-point theorem in cones.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

one-dimensional \(p\)-Laplacian; singular problem; positive solution; multiplicity; fixed-point index; fixed-point theorem; cone; superlinearity; sublinearity

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