Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders.
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Date
2017
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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.
Description
Keywords
Positive solution, Degenerated operator, Variational methods
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: <https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-017-0783-z>. Acesso em: 19 mar. 2019.