Spectral properties and nodal solutions for second-order, \(m\)-point, \(p\)-Laplacian boundary value problems
For \(p > 1\), the \(p\)-Laplacian \(m\)-point boundary value problem is considered: \[ -\phi_p(u')' = f(u)\text{ on }(0, 1), \] with the boundary conditions \[ u(0) = 0,\quad u(1) = \sum_{i=1}^{m-2} \alpha_iu(\eta_i), \] where \(\phi_p(s) := |s|^{p-1} sgn \;s\) for \(s \in \mathbb R\), \(m \geq 3\), \(\alpha_i, \eta_i \in (0, 1), i = 1, \dots, m-2\), and \(\sum_{i=1}^{m-2} \alpha_i < 1\). Also \(f: \mathbb R \rightarrow \mathbb R\) is continuous and assumed to satisfy \(sf(s) > 0\) for \(s \in \mathbb R \backslash \{ 0 \}\) as well as \(f_0 = \lim_{\xi \rightarrow 0} \frac{f(\xi)}{\phi_p(\xi)} > 0\), whereas a nodal solution means a sign-changing solution having a given number of zeros. Let \(X\) be the set of functions in \(\{ u \in C^1[0, 1]: \phi_p (u') \in C^1[0, 1] \}\) which further satisfy the boundary conditions as above mentioned. \(Y := C^0[0, 1]\). With the norm on \(X\) being \(\| \cdot \|_X := |u|_1 + |\phi_p(u')|_1\) and the norm on \(Y\) being \(\| \cdot \|_Y = | \cdot |_0\), consider the operator \(\bigtriangleup_p: X \longrightarrow Y\) defined by \(\bigtriangleup_p u = \phi_p(u')'\). First the spectral properties for this operator along with topological degree theoretic results are investigated. Then a Rabinowitz-type global bifurcation result for the above mentioned equation is derived, and used to obtain nodal solutions.
scientific article; zbMATH DE number 5548692