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- Title
A group-theoretical approach for nonlinear Schrödinger equations.
- Authors
Molica Bisci, Giovanni
- 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 ℝ d {\mathbb{R}^{d}} ( d ≥ 3 {d\geq 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) {O(d)}. In addition, if the nonlinear term is odd, and d > 3 {d>3} , the existence of (- 1) d + [ d - 3 2 ] {(-1)^{d}+[\frac{d-3}{2}]} pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the L ∞ {L^{\infty}} -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.
- Subjects
SCHRODINGER equation; NONLINEAR Schrodinger equation; SCHRODINGER operator
- Publication
Advances in Calculus of Variations, 2020, Vol 13, Issue 4, p403
- ISSN
1864-8258
- Publication type
Article
- DOI
10.1515/acv-2018-0016