8 problems

Let $P u=-\Delta u+V(x)u+u$ be subcritical in $\mathbb{R}^n$, and assume that … If $u$ is a positive solution of $Pu=0$ in $\mathbb{R}^n$ satisfying the stated condition, then let…

Full-parameter properties conjecture. The following assertions hold:

Aharonov–Bohm Hardy conjecture. If $\beta\notin\mathbb{Z}$, then there exists a constant $\lambda(p)>\left(\frac{2-p}{p}\right)^p$ such that

Let $d\geq 2$, let $\nabla_A$ denote the magnetic gradient, let $h_{A,p}$ be the corresponding magnetic $p$-Laplacian form domain, and let $\mu_{p,d}$ be the Hardy constant. Let…

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $N>k\geq 2$, and let $u\in C_c^k(B_R)$. Critical logarithmic Hardy conjecture. The inequality … holds. Moreover, the cons…

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $1<p<\infty$, let $k\geq 2$, and let $u\in C_c^k(B_R)$. Write $\operatorname{dist}(x,\partial B_R)$ for the distance from…

Let $B_R$ be the ball of radius $R$ centred at the origin in $\mathbb{R}^N$, let $k\geq 2$, let $1<p<N/k$, let $0<\gamma\leq (N-kp)/(kp-1)$, and let $u\in C_c^k(B_R)$. Weighted hig…

Let $\Omega$ be a globally twisted waveguide with twist derivative $\dot\theta=\beta+\varepsilon$, where $\beta$ is the constant twisting rate and $\varepsilon$ is a nonzero pertur…