Abstract weak Harnack inequality, multiple fixed points and \(p\)-Laplace equations
In this paper, the author establishes an existence and localization result for fixed points of a completely continuous operator \(N\) defined on a cone \(K_0\) of a Banach space \(X\). In this result, the key assumption is the abstract weak Harnack inequality \[ Nu\geq \|Nu\|\chi\quad\text{for every}\quad u\in K_0, \] where \(\chi \in K_0\setminus\{0\}\), \(\leq\) is the partial order relation associated to \(K_0\) and \(\|\cdot\|\) is a continuous seminorm which is monotone with respect to \(\leq\). With some additional technical assumptions on \(N\), the authors proves the existence of at least a fixed point for \(N\) belonging to a conical annular region of \(K_0\). Moreover, exploiting the localization property of the fixed points, the author obtains, in a natural way, the existence of multiple fixed points for the operator \(N\). The obtained abstract results are applied to the Dirichlet problem for nonlinear elliptic equations involving the \(p\)-Laplace operator. In particular, the author proves the existence of sequences of positive weak solutions assuming that the nonlinearity has a suitable oscillatory behavior.
scientific article; zbMATH DE number 6162216
scientific article