An energy constrained method for the existence of layered type solutions of NLS equations
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, p. 725-749

We study the existence of positive solutions on ${ℝ}^{N+1}$ to semilinear elliptic equation $-\Delta u+u=f\left(u\right)$ where $N⩾1$ and f is modeled on the power case $f\left(u\right)={|u|}^{p-1}u$. Denoting with c the mountain pass level of $V\left(u\right)=\frac{1}{2}{\parallel u\parallel }_{{H}^{1}\left({ℝ}^{N}\right)}^{2}-{\int }_{{ℝ}^{N}}F\left(u\right)\phantom{\rule{0.166667em}{0ex}}dx$, $u\in {H}^{1}\left({ℝ}^{N}\right)$ ($F\left(s\right)={\int }_{0}^{s}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt$), we show, via a new energy constrained variational argument, that for any $b\in \left[0,c\right)$ there exists a positive bounded solution ${v}_{b}\in {C}^{2}\left({ℝ}^{N+1}\right)$ such that ${E}_{{v}_{b}}\left(y\right)=\frac{1}{2}{\parallel {\partial }_{y}{v}_{b}\left(·,y\right)\parallel }_{{L}^{2}\left({ℝ}^{N}\right)}^{2}-V\left({v}_{b}\left(·,y\right)\right)=-b$ and $v\left(x,y\right)\to 0$ as $|x|\to +\infty$ uniformly with respect to $y\in ℝ$. We also characterize the monotonicity, symmetry and periodicity properties of ${v}_{b}$.

DOI : https://doi.org/10.1016/j.anihpc.2013.07.003
Classification:  35J60,  35B08,  35B40,  35J20,  34C37
Keywords: Semilinear elliptic equations, Locally compact case, Variational methods, Energy constraints
@article{AIHPC_2014__31_4_725_0,
author = {Alessio, Francesca and Montecchiari, Piero},
title = {An energy constrained method for the existence of layered type solutions of NLS equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {4},
year = {2014},
pages = {725-749},
doi = {10.1016/j.anihpc.2013.07.003},
zbl = {06349267},
mrnumber = {3249811},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_4_725_0}
}

Alessio, Francesca; Montecchiari, Piero. An energy constrained method for the existence of layered type solutions of NLS equations. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 4, pp. 725-749. doi : 10.1016/j.anihpc.2013.07.003. http://www.numdam.org/item/AIHPC_2014__31_4_725_0/

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