arXiv subjects

Xingyong Zhang

Publications and source records attributed to Xingyong Zhang.

30 records · Page 2Linked to original sources

arXiv2023

Existence of solutions for a poly-Laplacian system involving concave-convex nonlinearity on locally finite graphs

We investigate the existence of two nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities and parameters with Dirichlet boundary condition on locally finite graphs. By using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one nontrivial solution of positive energy and one nontrivial solution of negative energy, respectively. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result in [4] which is based on the fibering method and Nehari manifold, we obtain the existence of ground state solution to the single equation corresponding to poly-Laplacian system. Especially, we present some ranges of parameters in all of results.

math.AP↗

arXiv2023

Existence of ground state sign-changing solutions for a class of quasilinear scalar field equations originating from nonlinear optics

This paper is mainly concerned with the existence of ground state sign-changing solutions for a class of second order quasilinear elliptic equations in bounded domains which derived from nonlinear optics models. Combining a non-Nehari manifold method due to Tang and Cheng [31] and a quantitative deformation lemma with Miranda theorem, we obtain that the problem has at least one ground state sign-changing solution with two precise nodal domains. We also obtain that any weak solution of the problem has $C^{1,σ}$-regularity for some $σ\in(0,1)$. With the help of Maximum Principle, we reach the conclusion that the energy of the ground state sign-changing solutions is strictly larger than twice that of the ground state solutions.

math.AP↗

arXiv2023

Ground state sign-changing homoclinic solutions for a discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity

By using a direct non-Nehari manifold method from [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.], we obtain an existence result of ground state sign-changing homoclinic solution which only changes sign one times and ground state homoclinic solution for a class of discrete nonlinear $p$-Laplacian equation with logarithmic nonlinearity. Moreover, we prove that the sign-changing ground state energy is larger than twice of the ground state energy.

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*} (-Δ_Φ)^{s}u+V(x)a(|u|)u=f(x,u),\quad x\in\mathbb{R}^N, \end{eqnarray*} where $(-Δ_Φ)^{s}(s\in(0, 1))$ denotes the non-local and maybe non-homogeneous operator, the so-called fractional $Φ$-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 nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities

We investigate the multiplicity of solutions for a quasilinear scalar field equation with a nonhomogeneous differential operator defined by \begin{eqnarray} Su:=-\mbox{div}\left\{ϕ\left(\frac{u^{2}+|\nabla u|^{2}}{2}\right)\nabla u\right\}+ϕ\left (\frac{u^{2}+|\nabla u|^{2}}{2}\right)u, \end{eqnarray} where $ϕ:[0,+\infty)\rightarrow\mathbb{R}$ is a positive continuous function. This operator is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only $\nabla u$ but also $u$. This particular quasilinear term generally appears in the study of nonlinear optics model which describes the propagation of self-trapped beam in a cylindrical optical fiber made from a self-focusing dielectric material. When the reaction term is concave-convex nonlinearities, by using the Nehari manifold and doing a fine analysis associated on the fibering map, we obtain that the equation admits at least one positive energy solution and negative energy solution which is also the ground state solution of the equation. We overcome two main difficulties which are caused by the nonhomogeneity of the differential operator $S$: (i) the almost everywhere convergence of the gradient for the minimizing sequence $\{u_{n}\}$; (ii) seeking the reasonable restrictions about $S$.

math.AP↗

arXiv2023

Existence of three solutions for a poly-Laplacian system on graphs

We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a $(p,q)$-Laplacian system with a parameter on locally finite graphs.The main tool is an abstract critical points theorem in [Bonanno and Bisci, J.Math.Appl.Anal, 2011, 382(1): 1-8]. A key point in this paper is that we overcome the difficulty to prove that the G$\hat{\mbox{a}}$teaux derivative of the variational functional for poly-Laplacian operator admits a continuous inverse, which is caused by the special definition of the poly-Laplacian operator on graph and mutual coupling of two variables in system.

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↗

arXiv2023

Existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian system on locally finite graphs

We generalize two embedding theorems and investigate the existence and multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system with perturbations and two parameters $λ_1$ and $λ_2$ on locally finite graph. By using the Ekeland's variational principle, we obtain that system has at least one nontrivial solution when the nonlinear term satisfies the sub-$(p,q)$ conditions. We also obtain a necessary condition for the existence of semi-trivial solutions to the system. Moreover, by using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one solution of positive energy and one solution of negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions which is weaker than the well-known Ambrosetti-Rabinowitz condition. Especially, in all of the results, we present the concrete ranges of the parameters $λ_1$ and $λ_2$.

math.AP↗

arXiv2022

Existence and multiplicity of solutions for a class of Kirchhoff type $(Φ_1,Φ_2)$-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}(ϕ_1(|\nabla u|)\nabla u)+V_1(x)ϕ_1(|u|)u=λF_u(x, u,v), \ \ x\in \mathbb R^N, -\mbox{div}(ϕ_2(|\nabla v|)\nabla v)+V_2(x)ϕ_2(|v|)v=λF_v(x, u,v), \ \ x\in \mathbb R^N, u\in W^{1,Φ_1}(\mathbb R^N), v\in W^{1,Φ_2}(\mathbb R^N), \end{cases} \end{eqnarray*} where $N\ge 2$, $\inf_{\mathbb R^N}V_i(x)>0,i=1,2$, and $λ>0$. 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 $(u_λ,v_λ)$ with $\|(u_λ,v_λ)\|_{\infty}\le 2$ for every $λ$ large enough, and the families of solutions $\{(u_λ,v_λ)\}$ satisfy that $\|(u_λ,v_λ)\|\to 0$ as $λ\to \infty$. 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 $λ$. By mountain pass theorem, we obtain that system has at least one non-trivial weak solution $u_λ$ under some local superquadratic conditions for each given large parameter $λ$. We get a concrete lower bound of the parameter $λ$, and then obtain two estimates of weak solutions $u_λ$. We also obtain that $u_λ\to 0$ if $λ$ tends to $\infty$. Finally, we present an example as an application of our results.

math.AP↗

arXiv2016

Multiple periodic solutions for two classes of nonlinear difference systems involving classical $(ϕ_1,ϕ_2)$-Laplacian

In this paper, we investigate the existence of multiple periodic solutions for two classes of nonlinear difference systems involving $(ϕ_1,ϕ_2)$-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for the first nonlinear difference system with $(ϕ_1,ϕ_2)$-Laplacian and two parameters. Moreover, for the second nonlinear difference system with $(ϕ_1,ϕ_2)$-Laplacian, by using the Clark's Theorem, we obtain a multiplicity result of periodic solutions under a symmetric condition. Finally, two examples are given to verify our theorems.

math.DS↗