Spectral properties of \(p\)-Laplacian problems with Neumann and mixed-type multi-point boundary conditions
The author first investigates the spectral problem of the \(p\)-Laplacian eigenvalue problem \[ -\phi_p(u')'(x)=\lambda\phi_p(u)(x),\quad0<x<1 \] together with Dirichlet-type, Neumann-type or mixed boundary conditions. Here, \(\phi_p(s)=| s|^{p-2}s\) is the \(p\)-Laplacian operator (\(p>1\)) and \(\lambda\) corresponds to an eigenvalue of the problem. Then the degree of an associated operator is defined and computed. Several applications are provided in this paper. First, a global bifurcation theory is developed for the nonlinear problem \[ -\phi_p(u')'(x)=\lambda f(u)(x),\quad 0<x<1, \] where \(f: \mathbb{R}\rightarrow\mathbb{R}\) is continuous and satisfies the sign condition \(sf(s)>0\) for \(s\in\mathbb{R}^*\) both with the growth condition \[ 0<\lim_{s\to0}\frac{f(s)}{\phi_p(s)}<\infty. \] The author proves Rabinowitz-type global bifurcation results. Then under a non-resonance condition, the existence of a solution is obtained for the nonlinear equation \[ -\phi_p(u')'=f(u)+h \] and also the existence of nodal solutions in case \(h=0\) is obtained. The paper ends with some properties of the spectrum of the operator \(-\phi_p(u')'\).
scientific article; zbMATH DE number 5843311
scientific article