An energy constrained method for the existence of layered type solutions of NLS equations

We study the existence of positive solutions on RN+1RN+1 to semilinear elliptic equation −Δu+u=f(u)−Δu+u=f(u) where N⩾1N⩾1 and f   is modeled on the power case f(u)=|u|p−1uf(u)=|u|p−1u. Denoting with c   the mountain pass level of V(u)=12‖u‖H1(RN)2−∫RNF(u)dx, u∈H1(RN)u∈H1(RN) (F(s)=∫0sf(t)dt), we show, via a new energy constrained variational argument, that for any b∈[0,c)b∈[0,c) there exists a positive bounded solution vb∈C2(RN+1)vb∈C2(RN+1) such that Evb(y)=12‖∂yvb(⋅,y)‖L2(RN)2−V(vb(⋅,y))=−b and v(x,y)→0v(x,y)→0 as |x|→+∞|x|→+∞ uniformly with respect to y∈Ry∈R. We also characterize the monotonicity, symmetry and periodicity properties of vbvb.