Advances in Differential Equations

Symmetry and multiple solutions for certain quasilinear elliptic equations

Roberta Filippucci, Patrizia Pucci, and Csaba Varga

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Abstract

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 [1] and Bonnano and Candito [2]. The methods we present work also for different types of symmetrization, see Van Schaftingen [22].

Article information

Source
Adv. Differential Equations Volume 20, Number 7/8 (2015), 601-634.

Dates
First available in Project Euclid: 8 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1431115710

Mathematical Reviews number (MathSciNet)
MR3344612

Zentralblatt MATH identifier
1326.35144

Subjects
Primary: 35A15: Variational methods 35J15: Second-order elliptic equations 35J20: Variational methods for second-order elliptic equations 35J62: Quasilinear elliptic equations

Citation

Filippucci, Roberta; Pucci, Patrizia; Varga, Csaba. Symmetry and multiple solutions for certain quasilinear elliptic equations. Adv. Differential Equations 20 (2015), no. 7/8, 601--634. https://projecteuclid.org/euclid.ade/1431115710.


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