The \((p, q)\)Hardy Inequalities And Application To Nonlinear Eigenvalue Problems
Speaker 
Nirjan Biswas


When 
Nov 22, 2021
from 02:00 PM to 03:00 PM 
Where  zoom meet 
Add event to calendar 
vCal iCal 
Abstract: For \(p\) \(∈ (1, ∞)\), \(q ∈ (0, ∞)\), and an open set \(Ω\) in \(R\)^N with \(N\) ≥ 2, we look for various function spaces for the weight functions \(g \in L_{loc}^1\) (Ω) that satisfy the following weighted Hardy inequality:
\(∫︁_Ωg(x)u(x)^q dx ≤ C(︂∫︁_Ω∇u(x)^p dx)︂^{\frac{q}{p}}, ∀ u ∈ C1c(Ω)\),
where \(C\) is a generic positive constant that does not dependent on \(u\). The weight functions satisfying the above inequality is referred to as \((p,q)\)Hardy potentials. Depending on the values of \(N, p,q\) and the geometry of Ω , we provide various Lorentz spaces, LorentzZygmund spaces, and weighted Lebesgue spaces consisting of \((p, q)\)Hardy potentials.
As an application, for \(p,q \in (1, \infty)\), we extend the classes of \((p,q)\)Hardy potentials for which the following nonlinear partial differential equation admits nontrivial solutions: $$\Delta_p u = \lambda g u^{q2} u \mbox{ in }\ \Omega, u = 0 \mbox{ on } \ \partial \Omega,\tag{D}$$
where \(\lambda\) is a real parameter, \(\Delta_p u\) = \({\rm div}(∇ u^{p2} ∇ u)\) is the \(p\)Laplace operator. For \(q=p\), we provide a large class of weight functions for the existence of positive principal eigenvalue, closedness and the infinitude of the set of eigenvalues of \((D)\).