This paper discusses some existence results for a quasi-linear elliptic equation with Neumann boundary values of the form \[ \triangle_p u=f(\cdot,u), \] where \(\triangle_p\) is the p-Laplacian. The authors obtain nontrivial solutions of saddle point character under certain growth conditions on \(f\). Similar results have been known before [X. Wu and K.-K. Tan, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 7, 1334–1347 (2006; Zbl 1109.35047)] though under different assumptions on the right hand side \(f\). In particular, the authors remove one growth condition and replace it by a Carathéodory condition. Please note that this Carathéodory condition on \(f\) is explicitly stated on p. 546 but not in the theorems.
The proofs mainly use standard ideas from the theory of the p-Laplacian as well as from calculus of variations, in particular Sobolev embedding theorems.