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

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.


34B15 Nonlinear boundary value problems for ordinary differential equations
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