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

Alessio, Francesca ; Montecchiari, Piero

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 725-749 / Harvested from Numdam

Résumé

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}$.

Détail

Publié le : 2014-01-01

MR 3249811 | Zbl 06349267

DOI : https://doi.org/10.1016/j.anihpc.2013.07.003
Classification:  35J60,  35B08,  35B40,  35J20,  34C37

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