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# The $$(p, q)$$-Hardy Inequalities And Application To Nonlinear Eigenvalue Problems

Nirjan Biswas
 Speaker Nirjan Biswas Nov 22, 2021 from 02:00 PM to 03:00 PM zoom meet 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:

$$∫︁_Ω|g(x)||u(x)|^q dx ≤ C(︂∫︁_Ω|∇u(x)|^p dx)︂^{\frac{q}{p}}, ∀ u ∈ C1c(Ω)$$,

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, Lorentz-Zygmund 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: $$-\Delta_p u = \lambda g |u|^{q-2} u \mbox{ in }\ \Omega, u = 0 \mbox{ on } \ \partial \Omega,\tag{D}$$

where $$\lambda$$ is a real parameter, $$\Delta_p u$$ =  $${\rm div}(|∇ u|^{p-2} ∇ u)$$ 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 $$(D)$$.

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