On positive solutions of indefinite inhomogeneous Neumann boundary value problems (Q1769840)

In the present study the authors investigate the existence and multiplicity of positive solutions for the following class of inhomogeneous Neumann boundary value problems with indefinite nonlinearities \[ \begin{cases} -\Delta_p u-\lambda k(x)|u|^{p-2}u=K(x)|u|^{\gamma+2}u\quad & \text{in }{\mathcal M}\;\quad(1)\\ |\nabla u|^{p-2}\,\frac{\partial u}{\partial n}+d(x)|u|^{p-2}u=D(x)|u|^{q-2}u & \text{on }{\mathcal M},\quad(2)\end{cases} \] where \(\mathcal M\) is a smooth connected compact Riemannian manifold of the dimension \(n\geq 2\) with metric \(g\) and boundary \(\partial M\). \(\Delta_p\) and \(\nabla\), respectively, denote the \(p\)-Laplace-Beltrami operator and the gradient in the metric \(g\). The main goal of the authors is to show how a fibering scheme can be useful for (1)--(2), and how this scheme leads to new existence and multiplicity results.