We derive whole series of new integral inequalities of the Hardy-type, with non-conjugate exponents. First, we prove and discuss two equivalent general inequalities of such type, as well as their corresponding reverse inequalities. General results are then applied to special Hardy-type kernel and power weights. Also, some estimates of weight functions and constant factors are obtained. In parti...

where Ω ⊂ R(N ≥ 4) is an open bounded domain with smooth boundary, β > 0, 0 ∈ Ω, 0 ≤ s < 2, 2∗(s) := 2(N − s) N − 2 is the critical Hardy-Sobolev exponent and, when s = 0, 2∗(0) = 2N N − 2 is the critical Sobolev exponent, 0 ≤ μ < μ := (N − 2) 4 . In [1] A. Ferrero and F. Gazzola investigated the existence of nontrivial solutions for problem (1.1) with β = 1, s = 0. In [2] D. S. Kang and S. J. ...

In this article, we study the existence and multiplicity of nontrivial solutions for a class of Hamiltoniam systems with weights and nonlinearity involving the Hardy-Sobolev exponents. Results are proved using variational methods for strongly indefinite functionals.

Abstract In this article, we first apply the method of combining interpolation theorem and weak-type estimate developed in Chen et al. to derive Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using weighted Hardy inequality, further obtain Stein-Weiss For extremal problem corresponding presence double-weighted exponents not being necessarily nonnegative makes it impossi...

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡...

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nerahi manifold and by exploring some properties of the best Hardy-Sobolev constan...