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

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 \((D)\).