6 citations · 11 across the 19 of their papers we have counts for
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On the integral formula of the Jacobian determinant
Shibo Liu
It is known that the integral of the Jacobian determinant of a smooth map depends only on and this result leads to an analyt…
Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero
Shibo Liu, Kanishka Perera
We study the following zero-mass Schr{ö}dinger-Poisson-Slater equation \[ - Δu + \left( \frac{1}{4 π| x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2…
Multiple solutions for Schr{ö}dinger-Poisson-Slater equations with critical growth
Shibo Liu
We obtain multiple solutions for the zero mass Schr{ö}dinger-Poisson-Slater equation \[ - Δu + \left( \frac{1}{4 π| x |} \ast u^2 \right) u = λg (x) | u |^{p - 2} u + | u |^{6 - 2}…
Polynomial 2D Biharmonic Coordinates for High-order Cages
Shibo Liu, Ligang Liu, Xiao-Ming Fu
We derive closed-form expressions of biharmonic coordinates for 2D high-order cages, enabling the transformation of the input polynomial curves into polynomial curves of any order.…