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

Abstract

We study the existence of positive solutions on ${\mathbb{R}}^{N+1}$ to semilinear elliptic equation $-\mathrm{\Delta }u+u=f(u)$ where $N⩾1$ and $f$ is modeled on the power case $f(u)=|u{|}^{p-1}u$. Denoting with $c$ the mountain pass level of $V(u)=\frac{1}{2}\Vert u{\Vert }_{H^1({\mathbb{R}}^N)}^2-{\int }_{{\mathbb{R}}^N}F(u)dx$, $u\in H^1({\mathbb{R}}^N)$ ($F(s)={\int }_0^{s}f(t)dt$), 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 $E_{v_b}(y)=\frac{1}{2}\Vert {\partial }_y{v}_b(\cdot ,y){\Vert }_{L^2({\mathbb{R}}^N)}^2-V(v_b(\cdot ,y))=-b$ and $v(x,y)\to 0$ as $|x|\to +\infty$ uniformly with respect to $y\in \mathbb{R}$. We also characterize the monotonicity, symmetry and periodicity properties of $v_b$.