The authors deal with the Neumann boundary value problem
where is a positive function in , and are continuous in , is positive; is a positive parameter.
The solutions to (1) are the critical points of a functional of the form , , . Here, is a norm of and is a primitive of with respect to . It is shown that, under assumptions that very roughly speaking express an oscillating character of , problem (1) may have multiple solutions (infinitely many, or at least three, depending on the particular assumptions). The main tool in the proof are variants of the Ricceri variational principle as given in the paper of G. Bonanno and P. Candito [J. Differ. Equations 244, No. 12, 3031–3059 (2008; Zbl 1149.49007)].