Time-mappings and multiplicity of solutions for the one-dimensional \(p\)- Laplacian. (English) Zbl 0792.34021

Manásevich, Raúl; Zanolin, Fabio

Nonlinear Anal., Theory Methods Appl. 21, No. 4, 269-291 (1993).

The paper is concerned with the quasilinear Dirichlet boundary value problem (P) \((\varphi_ p (u'))'+f(t,u)=0\), \(u(a)=u(b)=0\) where \(\varphi_ p:\mathbb{R} \to \mathbb{R}\) is defined by \(\varphi_ p(s)=| s |^{p-2}s\), \(p>1\), and \(f:[a,b] \times \mathbb{R}_ +\to \mathbb{R}\) is an \(L^ 1\)-Carathéodory function. The authors discuss the existence and multiplicity of positive solutions to (P). The methods used include the study of an associated time-mapping function, the use of the Leray- Schauder degree, and maximum principles.

Reviewer: S.Aizicovici (Athens / Ohio)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

quasilinear Dirichlet boundary value problem; existence; multiplicity of positive solutions; associated time-mapping function; Leray-Schauder degree; maximum principles

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