| Abstract: |
| We consider nonlinear scalar field equations on unbounded domains, where lack of compactness due to translations prevents the direct application of classical variational methods. This difficulty can be overcome by exploiting symmetry.
A key idea is to replace symmetry assumptions with suitable symmetry constraints. Building on the planar framework, we use barycentric-type conditions to control the distribution of mass and prevent concentration at infinity.
Within this approach, we develop a variational scheme based on constrained min--max constructions, where the constraints encode higher-order symmetry properties of the associated measures. This allows us to recover compactness at the level of Palais--Smale sequences and to obtain nontrivial solutions without imposing radial symmetry.
We then analyze the structure of these symmetry constraints, showing how they can be characterized in terms of higher-order barycenters and how, in the planar case, they lead to optimal configurations concentrated on the vertices of regular polygons. We also discuss their extension to higher dimensions, where new geometric and algebraic features arise.
This perspective provides a unified framework yielding existence and multiplicity results beyond the classical or block-radial symmetric setting. |
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