Nontrivial solutions for a class of resonant pp-Laplacian Neumann problems

We consider a nonlinear Neumann problem driven by the pp-Laplacian differential operator with a Carathéodory reaction term. We assume that asymptotically at infinity resonance occurs with respect to the principal eigenvalue λ0=0λ0=0 (i.e., the reaction term is p−1p−1-sublinear near +∞+∞). Using variational methods based on the critical point theory and an alternative minimax characterization of the first nonzero eigenvalue λ1>0λ1>0, we show that the problem has a nontrivial smooth strong solution.