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Title: Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders.
Authors: Assunção, Ronaldo Brasileiro
Miyagaki, Olimpio Hiroshi
Rodrigues, Bruno Mendes
Keywords: Positive solution
Degenerated operator
Variational methods
Issue Date: 2017
Citation: ASSUNÇÃO, R. B.; MIYAGAKI, O. H.; RODRIGUES, B. M. Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders. Boundary Value Problems, v. 2017, n. 52, p. 1-11, mar./abr. 2017. Disponível em: <>. Acesso em: 19 mar. 2019.
Abstract: We consider the following elliptic problem: – div( |∇u| p–2∇u |y| ap ) = |u| q–2u |y| bq + f(x) in , u = 0 on ∂, in an unbounded cylindrical domain := (y,z) ∈ Rm+1 × RN–m–1;0< A < |y| < B < ∞ , where 1 ≤ m < N – p, q = q(a, b) := Np N–p(a+1–b) , p > 1 and A, B ∈ R+. Let p∗ N,m := p(N–m) N–m–p . We show that p∗ N,m is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Maz’ja’s inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev (p = 2, a = 0 and b = 0) and Hardy (p = 2, a = 0 and b = 1). Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case f ≡ 0 and at least two solutions in the case f ≡ 0, if p < q < p∗ N,m.
ISSN: 1687-2770
metadata.dc.rights.license: O periódico Boundary Value Problems permite o arquivamento da versão/PDF do editor no Repositório Institucional. Fonte: Sherpa/Romeo <>. Acesso em: 20 out 2016.
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