arXiv subjects

Cuiling Liu

Publications and source records attributed to Cuiling Liu.

6 recordsLinked to original sources

arXiv2025

Infinitely many solutions for an instantaneous and non-instantaneous fourth-order differential system with local assumptions

We investigate a class of fourth-order differential systems with instantaneous and non-instantaneous impulses. Our technical approach is mainly based on a variant of Clark's theorem without the global assumptions. Under locally subquadratic growth conditions imposed on the nonlinear terms $f_i(t,u)$ and impulsive terms $I_i$, combined with perturbations governed by arbitrary continuous functions of small coefficient $\varepsilon$, we establish the existence of multiple small solutions. Specifically, the system exhibits infinitely many solutions in the case where $\varepsilon=0$.

math.AP↗

arXiv2023

Ground state solutions for a non-local type problem in fractional Orlicz Sobolev spaces

In this paper, we study the following nonlocal problem in fractional Orlicz Sobolev spaces \begin{eqnarray*} (-\Delta_{\Phi})^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in\mathbb{R}^N, \end{eqnarray*} where $(-\Delta_{\Phi})^{s}(s\in(0, 1))$ denotes the non-local and maybe non-homogeneous operator, the so-called fractional $\Phi$-Laplacian. Without assuming the Ambrosetti-Rabinowitz type and the Nehari type conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case. The proof is based on a variant version of the mountain pass theorem and a Lions' type result for fractional Orlicz Sobolev spaces.

math.AP↗

arXiv2023

Multiplicity of solutions for a class of nonhomogeneous quasilinear elliptic system with locally symmetric condition in $\mathbb{R}^N$

This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole-space $\mathbb{R}^N$. By a variant of Clark's theorem without the global symmetric condition and a Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0 and the perturbation term is any continuous function with a small parameter and no any growth hypothesis.

math.AP↗

arXiv2022

Existence and multiplicity of solutions for a class of Kirchhoff type $(\Phi_1,\Phi_2)$-Laplacian system with locally super-linear condition in $\mathbb{R}^N$

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$ condition, that is, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in \mathbb{R}^N$. We obtain that system has at least one weak solution by using the classical Mountain Pass Theorem. To a certain extent, our theorems extend the results of Tang-Lin-Yu [Journal of Dynamics and Differential Equations, 2019, 31(1): 369-383]. Moreover, under the above naturally global restriction, we obtain that system has infinitely many weak solutions of high energy by using the Symmetric Mountain Pass Theorem, which is different from those results of Wang-Zhang-Fang [Journal of Nonlinear Sciences and Applications, 2017, 10(7): 3792-3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

math.AP↗

arXiv2020

Existence of solutions for a quasilinear elliptic system with local nonlinearity on $\mathbb R^N$

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system \begin{eqnarray*} \begin{cases}{ccc} -\mbox{div}(\phi_1(|\nabla u|)\nabla u)+V_1(x)\phi_1(|u|)u=\lambda F_u(x, u,v), \ \ x\in \mathbb R^N, -\mbox{div}(\phi_2(|\nabla v|)\nabla v)+V_2(x)\phi_2(|v|)v=\lambda F_v(x, u,v), \ \ x\in \mathbb R^N, u\in W^{1,\Phi_1}(\mathbb R^N), v\in W^{1,\Phi_2}(\mathbb R^N), \end{cases} \end{eqnarray*} where $N\ge 2$, $\inf_{\mathbb R^N}V_i(x)>0,i=1,2$, and $\lambda>0$. We obtain that when the nonlinear term $F$ satisfies some growth conditions only in a circle with center $0$ and radius $4$, system has a nontrivial solution $(u_\lambda,v_\lambda)$ with $\|(u_{\lambda},v_{\lambda})\|_{\infty}\le 2$ for every $\lambda$ large enough, and the families of solutions $\{(u_\lambda,v_\lambda)\}$ satisfy that $\|(u_\lambda,v_\lambda)\|\to 0$ as $\lambda\to \infty$. Moreover, a corresponding result for a quasilinear elliptic equation is also obtained, which is better than the result for the elliptic system.

math.AP↗

arXiv2019

Existence of solutions for Kirchhoff-type fractional Dirichlet problem with $p$-Laplacian

In this paper, we investigate the existence of solutions for a class of $p$-Laplacian fractional order Kirchhoff-type system with Riemann-Liouville fractional derivatives and a parameter $\lambda$. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_\lambda$ under some local superquadratic conditions for each given large parameter $\lambda$. We get a concrete lower bound of the parameter $\lambda$, and then obtain two estimates of weak solutions $u_\lambda$. We also obtain that $u_\lambda\to 0$ if $\lambda$ tends to $\infty$. Finally, we present an example as an application of our results.

math.AP↗