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

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