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

TL;DR

This paper introduces a new energy constrained variational method to establish the existence of layered positive solutions for nonlinear Schrödinger equations with power-type nonlinearities, analyzing their properties and behavior.

Contribution

The paper develops a novel energy constrained variational approach to prove existence and characterize layered solutions of NLS equations, extending previous methods.

Findings

Existence of positive layered solutions for all energy levels below the mountain pass level.

Solutions exhibit specific monotonicity, symmetry, and periodicity properties.

Solutions tend to zero uniformly as spatial variables tend to infinity.

Abstract

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 $v_{b}$.