In this article, we look for the weight functions (say g) that admits the following generalized Hardy-Rellich type inequality:for some constant C > 0, where Ω is an open set in R N with N ≥ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of D 2,2 0 (Ω) into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger. (2010): 35A23, 46E30, 46E35.
Mathematics Subject ClassificationSports. f * (t) := ess sup f, t = 0 inf{s > 0 : α f (s) < t}, t > 0.
We investigate the following eigenvalue problem, λ > 0 is a parameter, the weights L and K are measurable with L positive a.e. in A R2 R1 and K possibly sign-changing in A R2 R1 . We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. The asymptotic estimates for u(x) and ∇u(x) as |x| → R + 1 or R − 2 are also investigated.
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