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Shibo Liu

Publications and source records attributed to Shibo Liu.

At least 19 recordsLinked to original sources
arXiv2026

On biharmonic equations with $p$-Laplacian and indefinite potentials or critical nonlinearity

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.

math.AP↗
arXiv2026

Physical derivation of the coarea formula and an elementary proof via gradient flow

In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density $g (x)$. Then we present an rigorous proof using the changing variable formula. To this end we construct the diffeomorphism $\Phi$ via the gradient flow and compute its Jacobian determinant via geometric method.

math.GM↗
arXiv2026

ANVIL: Accelerator-Native Video Interpolation via Codec Motion Vector Priors

Real-time 30-to-60 fps video frame interpolation on mobile neural processing units (NPUs) requires each synthesized frame within 33.3 ms. We show that mainstream flow-based video frame interpolation faces three structural deployment barriers on mobile NPUs: spatial sampling operators exceed the frame budget or lack hardware support, iterative flow refinement collapses under 8-bit integer post-training quantization, and memory-bound operators dominate the inference graph. ANVIL addresses these barriers by reusing motion vectors from the H.264/AVC decoder to prealign input frames, removing learned optical flow, spatial sampling, and iterative accumulation from the accelerator graph. The remaining residual is refined by a convolution-dominated network composed almost entirely of compute-bound operators. On a Snapdragon 8 Gen 3 device, ANVIL achieves 12.8 ms 1080p inference at 8-bit integer precision; an open-source Android player sustains 28.4 ms median end-to-end latency over 30-minute continuous playback. Per-operator causal analysis identifies quantized accumulation on recurrent flow states as a key mechanism behind integer quantization failure in iterative methods. The current design targets H.264/AVC playback with decoder-exposed motion vectors.

eess.IV↗
arXiv2025

On the integral formula of the Jacobian determinant

It is known that the integral of the Jacobian determinant of a smooth map $f: \bar{\Omega} \rightarrow \mathbb{R}^n$ depends only on $f |_{\partial \Omega} $ and this result leads to an analytic proof of the Brouwer fixed point theorem. In this note we provide two new proofs of this result, one by classical analysis and one by differential forms and Stokes formula.

math.CA↗
arXiv2025

Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero

We study the following zero-mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] with nonlinearity subscaled near zero in the sense that $f (| x |, t) \approx a | t |^{p - 2} t$ as $| t | \rightarrow 0$ for some $p\in\big(\frac{18}{7},3\big)$. A nonzero solution is obtained via Morse theory when the nonlinearity is asymptotically scaled at infinity. For this purpose we prove an abstract result on the critical groups at infinity for functionals satisfying the geometric assumptions of the scaled saddle point theorem of Mercuri \& Perera [arXiv:2411.15887]. For the case that $f (| x |, \cdot)$ is odd, a sequence of solutions are obtained via a version of Clark's theorem due to Kajikiya [J.\ Funct.\ Anal.\ 225 (2005) 352--370].

math.AP↗
arXiv2025

Multiple solutions for Schr{\"o}dinger-Poisson-Slater equations with critical growth

We obtain multiple solutions for the zero mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = \lambda g (x) | u |^{p - 2} u + | u |^{6 - 2} u \text{, \ \ \ \ } u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] for $\lambda \gg 1$, where $p \in (4, 6)$ and $g \in L^{6 / (6 - p)} (\mathbb{R}^3)$. The crucial (PS)$_c$ condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.

math.AP↗
arXiv2025

Polynomial 2D Biharmonic Coordinates for High-order Cages

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. Central to our derivation is the use of the high-order boundary element method. We demonstrate the practicality and effectiveness of our method on various 2D deformations. In practice, users can easily manipulate the Bezier control points to perform the desired intuitive deformation, as the biharmonic coordinates provide an enriched deformation space and encourage the alignment between the boundary cage and its interior geometry.

cs.GR↗
arXiv2024

On $p(x)$-Laplacian equations in $\mathbb{R}^{N}$ with nonlinearity sublinear at zero

Let $p,q$ be functions on $\mathbb{R}^{N}$ satisfying $1\ll q\ll p\ll N$, we consider $p(x)$-Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert ^{q(x)-2}u+g(x,u)\text{,}\\ u\in W^{1,p(x)}(\mathbb{R}^{N})\text{.}% \end{array} \right. \] To apply variational methods, we introduce a subspace $X$ of $W^{1,p(x)}(\mathbb{R}^N)$ as our working space. Compact embedding from $X$ into $L^{q(x)}(\mathbb{R}^N)$ is proved, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to $\infty$ and $0$ respectively, when $g(x,\cdot)$ is odd.

math.AP↗
arXiv2024

Polynomial 2D Green Coordinates for High-order Cages

We propose conformal polynomial coordinates for 2D closed high-order cages, which consist of polynomial curves of any order. The coordinates enable the transformation of the input polynomial curves into polynomial curves of any order. We extend the classical 2D Green coordinates to define our coordinates, thereby leading to cage-aware conformal harmonic deformations. We extensively test our method on various 2D deformations, allowing users to manipulate the \Bezier control points to easily generate the desired deformation.

cs.CG↗
arXiv2023

Infinitely many solutions for Kirchhoff equations with indefinite potential

We obtain a sequence of solutions converging to zero for the Kirchhoff equation $$-\left( 1+\int_Ω\left\vert \nabla u\right\vert^2\right) Δu+V(x)u=f(u)\text{,\qquad}u\in H_{0}^{1}(Ω)$$ via truncating technique and a variant of Clark's theorem due to Liu--Wang, where $Ω$ is a bounded smooth domain $Ω\subset\mathbb{R}^N$. Similar result for Schrödinger-Poisson system on a bounded smooth domain $Ω\subset\mathbb{R}^3$ is also presented.

math.AP↗
arXiv2022

Quasilinear Schrödinger equations with concave and convex nonlinearities

In this paper, we consider the following quasilinear Schrödinger equation \begin{align*} -Δu-uΔ(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,} \end{align*} where $1 2\cdot2^{\ast}$. By taking advantage of geometric properties of a nonlinear transformation $f$ and a variant of Clark's theorem, we get a sequence of solutions with negative energy in a space smaller than $D^{1,2}(\mathbb{R}^{N})$. Nonnegative solution at negative energy level is also obtained.

math.AP↗
arXiv2019

On quasilinear elliptic problems with finite or infinite potential wells

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(ϕ(|\nabla u|)\nabla u\big)+V(x)ϕ(|u|)u=f(u)\qquad u\in W^{1,Φ}(\mathbb{R}^{N}), \] where $ϕ$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,Φ}(\mathbb{R}^{N})$ into $L^Φ(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=λa(x) + 1$ exhibits a steep potential well controlled by a positive parameter $λ$, we get nontrivial solutions for large $λ$.

math.AP↗
arXiv2019

On the Schrödinger-Poisson system with indefinite potential and $3$-sublinear nonlinearity

We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schrödinger operator $-Δ+V$ is indefinite. Previous results in this framework required $f$ to be strictly $3$-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where $f(t)=|t|^{2}t$; in this paper we fill this gap, obtaining non-trivial solutions when $f$ is not necessarily $3$-superlinear.

math.AP↗
arXiv2018

Standing waves for quasilinear Schröinger equations with indefinite potentials

We consider quasilinear Schrödinger equations in $\mathbb{R}^{N}$ of the form% \[ -Δu+V(x)u-uΔ(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schrödinger operator $-Δ+V$ is allowed to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that $g$ is odd, we get an unbounded sequence of solutions.

math.AP↗