JournalsaihpcVol. 31, No. 4pp. 725–749

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

  • Francesca Alessio

    Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, I 60131 Ancona, Italy
  • Piero Montecchiari

    Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, I 60131 Ancona, Italy
An energy constrained method for the existence of layered type solutions of NLS equations cover
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Abstract

We study the existence of positive solutions on RN+1\mathbb{R}^{N + 1} to semilinear elliptic equation Δu+u=f(u)−\mathrm{\Delta }u + u = f(u) where N1N⩾1 and f is modeled on the power case f(u)=up1uf(u) = |u|^{p−1}u. Denoting with c the mountain pass level of V(u)=12uH1(RN)2RNF(u)dxV(u) = \frac{1}{2}\|u\|_{H^{1}(\mathbb{R}^{N})}^{2}−\int _{\mathbb{R}^{N}}F(u)\:dx, uH1(RN)u \in H^{1}(\mathbb{R}^{N}) (F(s)=0sf(t)dtF(s) = \int _{0}^{s}f(t)\:dt), we show, via a new energy constrained variational argument, that for any b[0,c)b \in [0,c) there exists a positive bounded solution vbC2(RN+1)v_{b} \in C^{2}(\mathbb{R}^{N + 1}) such that Evb(y)=12yvb(,y)L2(RN)2V(vb(,y))=bE_{v_{b}}(y) = \frac{1}{2}\|\partial _{y}v_{b}( \cdot ,y)\|_{L^{2}(\mathbb{R}^{N})}^{2}−V(v_{b}( \cdot ,y)) = −b and v(x,y)0v(x,y)\rightarrow 0 as x+|x|\rightarrow + \infty uniformly with respect to yRy \in \mathbb{R}. We also characterize the monotonicity, symmetry and periodicity properties of vbv_{b}.

Cite this article

Francesca Alessio, Piero Montecchiari, An energy constrained method for the existence of layered type solutions of NLS equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 4, pp. 725–749

DOI 10.1016/J.ANIHPC.2013.07.003