# Non-coercive radially symmetric variational problems: Existence, symmetry and convexity of minimizers

## TL;DR

This paper establishes the existence and symmetry of solutions for a class of variational problems where traditional methods fail, using novel perturbation techniques to analyze minimizers.

## Contribution

It introduces a new approach employing superlinear perturbations to prove existence and properties of minimizers in challenging variational problems.

## Key findings

- Existence of radially symmetric solutions confirmed.
- Euler-Lagrange conditions validated for these solutions.
- Method applicable when standard calculus of variations techniques do not work.

## Abstract

We prove existence of radially symmetric solutions and validity of Euler-Lagrange necessary conditions for a class of variational problems such that neither direct methods nor indirect methods of Calculus of Variations apply. We obtain existence and qualitative properties of the solutions by means of ad-hoc superlinear perturbations of the functional having the same minimizers of the original one.