On positive solutions of indefinite inhomogeneous Neumann boundary value problems. (English) Zbl 1125.35358

Il’yasov, Yavdat; Runst, Thomas

Topol. Methods Nonlinear Anal. 24, No. 1, 41-67 (2004).

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.

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35J70	Degenerate elliptic equations
35J65	Nonlinear boundary value problems for linear elliptic equations
47J30	Variational methods involving nonlinear operators

Keywords:

multiplicity; variational approach; indefinite nonlinearities; fibering scheme