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EXISTENCE OF SOLUTION FOR TWO CLASSES OF QUASILINEAR SYSTEMS DEFINED ON A NON-REFLEXIVE ORLICZ-SOBOLEV SPACES
  • LUCAS DA SILVA,
  • Marco A.S. Souto
LUCAS DA SILVA
Universidade Federal de Campina Grande

Corresponding Author:ls3@academico.ufpb.br

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Marco A.S. Souto
Universidade Federal de Campina Grande
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Abstract

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type { − ∆ Φ 1 u = F u ( x , u , v )+ λ R u ( x , u , v ) in Ω − ∆ Φ 2 v = − F v ( x , u , v ) − λ R v ( x , u , v ) in Ω u = v = 0 on ∂ Ω where λ>0 is a parameter, Ω is a bounded domain in R N ( N≥2) with smooth boundary Ω. The first class we drop the ∆ 2 -condition of the functions Φ ̵̃ i ( i=1 ,2) and assume that F has a double criticality. For this class, we use a linking theorem without the Palais-Smale condition for locally Lipschitz functionals combined with a concentration–compactness lemma for nonreflexive Orlicz-Sobolev space. The second class, we relax the ∆ 2 -condition of the functions Φ i ( i=1 ,2). For this class, we consider F=0 and λ=1 and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Fréchet differentiable combined with some properties of the weak topology.
30 Apr 2024Submission Checks Completed
30 Apr 2024Assigned to Editor
07 May 2024Review(s) Completed, Editorial Evaluation Pending
17 May 2024Reviewer(s) Assigned