Pairs of positive solutions for the one-dimensional \(p\)-Laplacian. (English) Zbl 0813.34021

De Coster, C.

Nonlinear Anal., Theory Methods Appl. 23, No. 5, 669-681 (1994).

The paper studies the boundary value problem (1) \((\varphi_ p (u'))' + f(t,u) = 0\), \(u(0) = u(T) = 0\), where \(f\) is a Carathéodory function and \(\varphi_ p(s) = | s |^{p - 1} s\), \(p > 1\), (or more generally with an increasing homeomorphism \(\varphi : \mathbb{R} \to \mathbb{R}\) satisfying \(\varphi (0) = 0)\). The author proves the existence of at least two positive \(W^{2,1}\)-solutions of the problem. The argument is based on the Leray-Schauder degree theory and the upper and lower solutions method. The results are proved for example under the assumption that the behaviour of \(f\) near 0 and near \(\infty\) is controlled from a growth condition \(f(t,u) \geq b(t) \varphi_ p (u)\), where \(b\) is a positive \(L^ \infty\)-function, that \(\lambda_ 1 < 1\), where \(\lambda_ 1\) is the first eigenvalue of the spectral problem \((\varphi_ p (u'))' + \lambda b(t) \varphi_ p (u) = 0\), \(u(0) = u(T) = 0\), and that there exists a certain class of upper solutions of (1).

Reviewer: I.Rachůnková (Olomouc)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

positive solutions; \(p\)-Laplacian; multiplicity results; boundary value problem; Leray-Schauder degree theory; upper and lower solutions

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