  In this paper we study multiplicity and qualitative behavior of solutions for
semilinear elliptic problems with neumann boundary condition and asymptotically
linear smooth nonlinearity. We provide sufficient conditions on the number of
eigenvalues the derivative of the nonlinearity crosses to guarantee existence
of at least five nontrivial solutions. The techniques we use are a combination
of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la
Castro-Lazer.
