This paper is concerned with the existence of normalized solutions of the nonlinear Schr"odinger equation [ -Delta u+V(x)u+lambda u = |u|^{p-2}u qquad ext{in $mathbb{R}^N$} ] in the mass supercritical and Sobolev subcritical case $2+rac{4}{N}<2^*$. We prove the existence of a solution $(u,lambda)in H^1(mathbb{R}^N) imesmathbb{R}^+$ with prescribed $L^2$-norm $|u|_2= ho$ under various conditions on the potential $V:mathbb{R}^N omathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.

Bartsch, T., Molle, R., Rizzi, M., Verzini, G. (2020). Normalized solutions of mass supercritical Schrödinger equations with potential. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS.

Normalized solutions of mass supercritical Schrödinger equations with potential

Molle, R;
2020-08-17

Abstract

This paper is concerned with the existence of normalized solutions of the nonlinear Schr"odinger equation [ -Delta u+V(x)u+lambda u = |u|^{p-2}u qquad ext{in $mathbb{R}^N$} ] in the mass supercritical and Sobolev subcritical case $2+rac{4}{N}<2^*$. We prove the existence of a solution $(u,lambda)in H^1(mathbb{R}^N) imesmathbb{R}^+$ with prescribed $L^2$-norm $|u|_2= ho$ under various conditions on the potential $V:mathbb{R}^N omathbb{R}$, positive and vanishing at infinity, including potentials with singularities. The proof is based on a new min-max argument.
17-ago-2020
In corso di stampa
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
nonlinear Schrödinger equations; normalized solution; min-max methods
This work was supported by the MIUR Excellence Department Project CUP E83C18000100006 (Roma Tor Vergata University) and by the INdAM-GNAMPA group. M.R. supported by the Alexander von Humboldt foundation.
http://arxiv.org/abs/2008.07431v1
Bartsch, T., Molle, R., Rizzi, M., Verzini, G. (2020). Normalized solutions of mass supercritical Schrödinger equations with potential. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS.
Bartsch, T; Molle, R; Rizzi, M; Verzini, G
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/264612
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