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A group-theoretical approach for nonlinear Schrödinger equations

Giovanni Molica Bisci

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 d (d3). 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.


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

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