Multiple positive solutions for one dimensional third order \(p\)-Laplacian equations with integral boundary conditions
This paper is devoted to the following third order equation coupled to integral boundary conditions: \[ \left\{\begin{array}{l} \left(\Phi_{p}\left(u^{\prime \prime}\right)\right)^{\prime}+h(t) f(t, u(t))=0, \quad t \in(0,1), \\ u(0)-\alpha \,u^{\prime}(0)=\int_{0}^{1} g_{1}(s) u(s) d s, \\ u(1)+\beta \,u^{\prime}(1)=\int_{0}^{1} g_{2}(s) u(s) d s, \quad u^{\prime \prime}(0)=0. \end{array}\right. \] Here \(\alpha, \beta \geq 0\), \(p>1\), \(\Phi_{p}(u)=|u|^{p-2} u\) is called the one-dimensional \(p\)-Laplacian operator and \( \Phi_{p}^{-1}(u)=\Phi_{q}(u)=|u|^{q-2} u\) with \(\frac{1}{p}+\frac{1}{q}=1.\) Under suitable assumptions on the function \(f\), the authors deduce the existence of at least three positive solutions of the considered problem. The results are obtained from Avery-Peterson fixed point theorem. The used arguments consist on the construction of the Green's function related to the linear problem \[ u''(t)+y(t)=0, \; t \in(0,1), \quad u(0)-\alpha\, u^{\prime}(0)=0, \; u(1)+\beta\, u^{\prime}(1)=0, \] and recursive methods for kernel functions.
one-dimensional third-order \(p\)-Laplacian equations
scientific article; zbMATH DE number 7490422