We present some symmetrization results which we apply to the same abstract eigenvalue problem in order to show the existence of three different solutions which are invariant by Schwarz symmetrization. In particular, we introduce two different methods in order to prove the existence of multiple symmetric solutions. The first is based on the symmetric version of the Ekeland variational principle and the mountain pass theorem, while the latter consists of an application of a suitable symmetric version of the three critical points theorem due to Pucci and Serrin [17, 18], see Theorem 2.13 and its Corollary 2.14. Using the second method, we are able to improve some recent results of Arcoya and Carmona  and Bonnano and Candito . The methods we present work also for different types of symmetrization, see Van Schaftingen .
"Symmetry and multiple solutions for certain quasilinear elliptic equations." Adv. Differential Equations 20 (7/8) 601 - 634, July/August 2015. https://doi.org/10.57262/ade/1431115710