Our main goal is to prove the existence of multiple solutions with precise sign information for a Neumann problem driven by the $p$-Laplacian differential operator with a ($p-1$)-superlinear term which does not satisfy the Ambrosetti--Rabinowitz condition. Using minimax methods we show that the problem has five nontrivial smooth solutions, two positive, two negative and the fifth nodal. In the semilinear case ($p = 2$), using Morse theory, we produce a second nodal solution (for a total of six nontrivial smooth solutions).
"Multiple solutions for superlinear $p$-Laplacian Neumann problems." Osaka J. Math. 49 (3) 699 - 740, September 2012.