Nonlinear noncoercive Neumann problems with a reaction concave near the origin

We consider a nonlinear Neumann problem driven by the pLaplacianand with a concave parametric reaction term and an asymptotically linear pertur-bation. We prove a multiplicity theorem producing five nontrivial solutions all withsign information when the parameter is small. For the semilinear case (p = 2) weproduce six solutions, but we are unable to determine the sign of the sixth solution.Our approach uses critical point theory, truncation and comparison techniques andMorse theory.