Existence results for a problems involving Hardy Potentials
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Date
2024-06-10
Journal Title
Journal ISSN
Volume Title
Publisher
Mohamed Boudiaf University of M’sila, Faculty of Mathematics and Informatics, Departement of Mathematics
Abstract
The objective of our work is to study the Hardy inequality, Then we try to apply it to
the following elliptic linear problem
−∆u = γ
u
x2 + f(x) in Ω
u = 0 on ∂Ω
where f ∈ L
2
(Ω), 0 ∈ Ω, γ is a real parameter.
To prove that it admits a unique weak solution by using Lax Milligram theorem.
Then we have the following some elliptic problems involving Hardy potential
−∆u = γ
u
|x|
2 + f(x,u) in Ω
u = 0 on ∂Ω
where Ω ⊂ R
N (N > 2) be open and bounded, 0 ∈ Ω,γ is a real parameter.
Where we study the existence at least non-trivial weak solution using Mountain-Pass
theorem, that is the associated functional Jγ admits at least a non trivial critical point.
We present the following a class of Kirchhoff type problem involving Hardy type potentials
−M(
R
Ω |∇u|
2
dx)∆u =
µ
x2 a(x)u + λf(x,u) in Ω
u = 0 on ∂Ω
where Ω ⊂ R
N (N ≥ 3) is bounded domain with smooth boundary ∂Ω, 0 ∈ Ω,
M : R
+
0 → R is continuous and increasing function with R
+
0
:= [0,+∞), the function
a : Ω → R may change sign, λ is positive parameter,0 ≤ µ < 1
CN,2
, where CN,2 =
2
N−2
2
is optimal constant in the Hardy Inequality.
Description
Keywords
Hardy potentials, Variational methods, Critical point, Weak solution, mountain pass theorem., Kirchhoff type problem, Hardy type potential, Sub-linear non-linearity, Multiple solutions, Three critical points theorem