6 citations · 11 across the 19 of their papers we have counts for
20 papers
On biharmonic equations with -Laplacian and indefinite potentials or critical nonlinearity
Shibo Liu, Kanishka Perera
In this paper we consider nonlinear biharmonic equations with -Laplacian () of the form $$ \left\{ \begin{array}{l} Δ^2 u - Δ_p u + V (x) u = f (x, u) \text{,} u \in H^2…
Physical derivation of the coarea formula and an elementary proof via gradient flow
Shibo Liu
In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density . Then we present an rigorous proof using the changi…
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.…
On -Laplacian equations in with nonlinearity sublinear at zero
Shibo Liu, Chunshan Zhao
Let be functions on satisfying , we consider -Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -Δ_{p(x)}u+V(x)\vert u\…
Polynomial 2D Green Coordinates for High-order Cages
Shibo Liu, Ligang Liu, Xiao-Ming Fu
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…
Infinitely many solutions for Kirchhoff equations with indefinite potential
Shuai Jiang, Shibo Liu
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}(…
Standing waves for 6-superlinear Chern-Simons-Schrödinger systems with indefinite potentials
Shuai Jiang, Shibo Liu
In this paper we consider 6-superlinear Chern-Simons-Schrödinger systems. In contrast to most studies, we consider the case where the potential is indefinite so that the Schröd…
Quasilinear Schrödinger equations with concave and convex nonlinearities
Shibo Liu, Li-Feng Yin
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{…
On quasilinear elliptic problems with finite or infinite potential wells
Shibo Liu
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 and $f…
On the Schrödinger-Poisson system with indefinite potential and -sublinear nonlinearity
Sunra J. N. Mosconi, Shibo Liu
We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity and coercive sign-changing potential so that the Schrödinger operator is…
Solutions for quasilinear fourth order elliptic equations on with sign-changing potential
Shibo Liu, Zhihan Zhao
We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on with sign-changing potential. Our results generalize some recent re…
On the existence of vectors dual to a set of linear functionals
Shibo Liu
We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].
Standing waves for quasilinear Schröinger equations with indefinite potentials
Shibo Liu, Jian Zhou
We consider quasilinear Schrödinger equations in of the form% \[ -Δu+V(x)u-uΔ(u^{2})=g(u)\text{,}% \] where is -superlinear. Unlike all known results in…
On the change of variables formula for multiple integrals
Shibo Liu, Yashan Zhang
In this paper, we develop an elementary proof of the change of variables in multiple integrals. Our proof is based on an induction argument. Assuming the formula for (m-1)-integral…
Existence results for fractional p-Laplacian problems via Morse theory
Antonio Iannizzotto, Shibo Liu, Kanishka Perera +1
We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of con…
Standing waves with large frequency for 4-superlinear Schrödinger-Poisson systems
Huayang Chen, Shibo Liu
We consider standing waves with frequency for 4-superlinear Schrödinger-Poisson system. For large the problem reduces to a system of elliptic equations in wi…
Homology of saddle point reduction and applications to resonant elliptic systems
Chong Li, Shibo Liu
In the setting of saddle point reduction, we prove that the critical groups of the original functional and the reduced functional are isomorphic. As application, we obtain two nont…