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

We study the existence of positive solutions on ${\mathbb{R}}^{N+1}$ to semilinear elliptic equation $-\mathrm{\Delta }u+u=f(u)$ where $N\ge 1$ and $f$ is modeled on the power case $f(u)=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$. Denoting with $c$ the mountain pass level of $\text{f}(u)=\frac{1}{2}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1({\mathbb{R}}^N)}^2-{\int }_{{\mathbb{R}}^N}F(u)dx$, $u\in H^{1}(\R^{N})$($F(s)=\int_{0}^{s}f(t), dt$), we show, via a new energy constrained variational argument, that for any $b\in [0,c)$ there exists a positive bounded solution $v_b\in C^2({\mathbb{R}}^{N+1})$ such that $E_{v_{b}}(y)=\tfrac 12|\partial_{y}v_{b}(\cdot,y)|^{2}{L^{2}(\R^{N})}-V(v{b}(\cdot,y))=-b$and$v(x,y)\to 0$as$|x|\to+\infty$uniformlywithrespectto$y\in\R$. We also characterize the monotonicity, symmetry and periodicity properties of $v_b$.