A group-theoretical approach for nonlinear Schrödinger equations

Giovanni Molica Bisci

doi:10.1515/acv-2018-0016

Published by De Gruyter July 4, 2018

A group-theoretical approach for nonlinear Schrödinger equations

Giovanni Molica Bisci

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2018-0016

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Abstract

The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space ℝd (d≥3). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of the orthogonal group O⁢(d). In addition, if the nonlinear term is odd, and d>3, the existence of (-1)d+[d-32] pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the L∞-positive and radially symmetric potential term W governing the Schrödinger equations is requested. A concrete example of an application is pointed out. Finally, we emphasize that the method adopted here should be applied for a wider class of energies largely studied in the current literature also in non-Euclidean setting as, for instance, concave-convex nonlinearities on Cartan–Hadamard manifolds with poles.

Keywords: Schrödinger equation; variational methods; principle of symmetric criticality; radial and non-radial solutions

MSC 2010: 35J91; 35J60; 35A01; 45A15

Communicated by Giuseppe Mingione

Funding statement: The paper is realized with the auspices of the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009) and the INdAM-GNAMPA Project 2017 titled Teoria e modelli non-locali.

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Received: 2018-03-16

Accepted: 2018-06-05

Published Online: 2018-07-04

Published in Print: 2020-10-01

A group-theoretical approach for nonlinear Schrödinger equations

Abstract

References

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