Existence and multiplicity of solutions for Schrödinger equations with asymptotically linear nonlinearities allowing interaction with essential spectrum

We study the nonlinear problem -Delta u + V (x)u = f (u), x is an element of R-N, lim(|x|->infinity)u(x) = 0, where the Schr & ouml;dinger operator -Delta+V is positive and f is asymptotically linear. Moreover, lim(|x|->infinity )V(x) = sigma(0). We allow the interference of essential spectrum, i.e. sup(t not equal 0) f(t) / t >= sigma(0). If sup(t not equal 0 )2F(t)/t(2) <sigma(0), the existence of four solutions will be proved by Morse theory. If sup(t not equal 0 )2F(t)/t(2 )>= sigma(0), we can find a positive solution when mes({x is an element of R-N : V(x) > sigma(0)}) > 0.