Existence results for a problems involving Hardy Potentials

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.