Energy bound for sign changing solutions of an asymptotically linear elliptic equation in RNRN

In this paper, we give an estimate of a lower bound for the energy of sign changing solutions of the following nonlinear elliptic equation equation(∗ )−Δu+u=f(u),u∈H1(RN), where ff is locally Lipschitz continuous and asymptotically linear.Weth (2006) [5] gave a lower bound for the energy of sign changing solutions of Eq. (∗) if ff is superlinear. Here, we show that the result of Weth (2006) [5] is also true if ff is asymptotically linear.The new ingredient of this problem is to prove that the energy of a mountain pass solution is equal to the ground state energy (see Lemma 2.1), which is obvious when ff is superlinear but nontrivial when ff is asymptotically linear.