Three solutions for a mixed boundary value problem involving the one-dimensional \(p\)-Laplacian
This paper deals with two mixed boundary value problems of the form \[ -(| u'| ^{p-2}u')'=\lambda f(t,u), \quad u(a)=0=u'(b), \] and \[ -(| u'| ^{p-2}u')'+| u| ^{p-2}u=\lambda f(t,u), \quad u(a)=0=u'(b), \] where \(f\) is \(L^1\)-Carathéodory, \(p>1\) and \(\lambda >0\). Sufficient conditions are given which guarantee the existence of at least three solutions. For example, the following corollary is proved: ``Let \(f\) be positive, bounded and such that \(8 \int_0^{1/2} f(\xi) d\xi < {1 \over 3} \int_0^1 f(\xi) d\xi\). Then the problem \(-u''= f(u), \quad u(0)=0=u'(1)\), has at least three positive solutions.'' The proof is performed by applying an abstract three critical point theorem due to the first author and \textit{G. Bonanno} [Topol. Methods Nonlinear Anal. 22, No. 1, 93--103 (2003; Zbl 1048.58005)].
scientific article; zbMATH DE number 2117739