Complete structure of the Fučík spectrum of the \(p\)-Laplacian with integrable potentials on an interval (Q2828654)

Fučík spectrum has been widely used to study boundary value problems and Lagrangian stability of semilinear equations. In this article the authors investigate the \(p\)-Laplacian with integrable potentials on unit interval under the Dirichlet ( \(x(0)=x(1)=0\)) or the Neumann (\(x'(0)=x'(1)=0\)) boundary conditions.NEWLINENEWLINEThe main results of the article are the following: i) the article includes complete characterization of these Fučík spectra: each of them is composed of one horizontal line, one vertical line and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; ii) all asymptotic lines of these spectral curves are precisely described by using (Sturm-Liouville) eigenvalues of the \(p\)-Laplacian with potentials; iii) all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent in a certain sense on potentials.NEWLINENEWLINEThe article consists of four sections, Section 2 includes some preliminary results, Sections 3 and 4 treat the Fučík spectrum in the case of Dirichlet and the Neumann boundary conditions respectively.