A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$

In the paper we prove the weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int _{\mathbb {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$ ∫ R N V φ 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + K ∫ R N φ 2 μ ( x ) d x , for functions $$\varphi $$ φ in a weighted Sobolev space $$H^1_\mu $$ H μ 1 , for a wider class of potentials V than inverse square potentials and for weight functions $$\mu $$ μ of a quite general type. The case $$\mu =1$$ μ = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators $$\begin{aligned} Lu=\varDelta u+\frac{\nabla \mu }{\mu }\cdot \nabla u \end{aligned}$$ L u = Δ u + ∇ μ μ · ∇ u perturbed by singular potentials.

The Existence of Positive Solution for Semilinear Elliptic Equations with Multiple an Inverse Square Potential and Hardy-Sobolev Critical Exponents

Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.