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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.

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We study positive solutions to the boundary value problem $$ \displaylines{ -\Delta_p u - \Delta_q u = \lambda f(u) \quad \text{in } \Omega, \cr u = 0 \quad \text{on } \partial\Omega, }$$ where \(q \in (1,p)\) and &Omega; is a bounded domain in \(\mathbb{R}^N\), \(N>1\) with smooth boundary, \(\lambda\) is a positive parameter, and \(f:[0,\infty) \to (0,\infty)\) is \(C_1\), nondecreasing, and p-sublinear at infinity i.e. \(\lim_{t \to \infty} f(t)/t^{p-1}=0\). We discuss existence and multiplicity results for classes of such f. Further, when N=1, we discuss an example which exhibits S-shaped bifurcation curves. For more information see  https://ejde.math.txstate.edu/special/01/a3/abstr.html

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