Searcharxiv⌕ Search

arXiv subjects

Xingyong Zhang

Publications and source records attributed to Xingyong Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Existence of a nonnegative bound state for a higher-order logarithmic Schrödinger equation

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -αΔu+V(x)u-βu\ln|u| = γu(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $α,β,γ>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a constant at infinity. The simultaneous presence of the first- and higher-order logarithmic terms leads to a nonsmooth variational structure and additional compactness difficulties. We introduce a family of power-law approximations and derive estimates that are uniform as the approximation exponent tends to the logarithmic limit. Under a structural condition on the autonomous nonlinearity, we obtain uniqueness, up to translations, of the nonnegative autonomous profile with connected positivity set and a corresponding energy gap below the two-profile threshold. The small-amplitude behavior exhibits a maximum-principle/compact-support dichotomy: the case $m=1$ yields positivity, whereas the higher odd orders fall into the compact-support regime for the autonomous profile. We then establish a profile decomposition for constrained Palais-Smale sequences and construct a barycenter-based min-max level below the splitting threshold. This prevents loss of mass through multiple profiles and allows us to pass to the logarithmic limit. Under the stated structural and energy conditions, we obtain a nontrivial nonnegative bound state of the original equation.

math.AP↗
arXiv2026

Three solutions for a quasilinear inclusion systems driven by a nonstandard Laplacian operator in $\mathbb{R}^N$

This paper establishes the existence of three distinct weak solutions for a class of nonhomogeneous quasilinear inclusion systems, which are governed by locally Lipschitz functionals within Orlicz-Sobolev spaces over unbounded domains $\mathbb{R}^{N}$.By employing a new three critical points theorem established by Wu-Zhou in [29], we extend the nonsmooth critical point theory to handle scenarios with nonlinear growth conditions.Our analysis imposes a precise structural condition on the nonlinear term $H$, requiring it to exhibit superlinear growth while maintaining subcritical asymptotic behavior. This condition establishes a delicate balance between growth restrictions and functional analytic requirements. The novelties of this work include the construction of suitable energy functionals that are consistent with both the structure of Orlicz-Sobolev spaces and the characteristics of nonlinear growth. Through rigorous variational analysis and careful estimation techniques, we establish the existence of three weak solutions under these generalized conditions, thereby significantly expanding the applicability of critical point methods in nonstandard function spaces.

math.AP↗
arXiv2025

Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type

In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with $(p,q)$-Kirchhoff type \begin{eqnarray*} \begin{cases} -\Big[M_1\left(\int_Ω|\nabla u_1|^p dx\right)\Big]^{p-1}Δ_p u_1+\Big[M_3\left(\int_Ωa_1(x)|u_1|^p dx\right)\Big]^{p-1}a_1(x)|u_1|^{p-2}u_1=G_{u_1}(x,u_1,u_2)\ \ \mbox{in }Ω, -\Big[M_2\left(\int_Ω|\nabla u_2|^q dx\right)\Big]^{q-1}Δ_q u_2+\Big[M_4\left(\int_Ωa_2(x)|u_2|^q dx\right)\Big]^{q-1}a_2(x)|u_2|^{q-2}u_2=G_{u_2}(x,u_1,u_2)\ \ \mbox{in }Ω, u_1=u_2=0\ \ \quad \quad \quad \quad \quad \quad \quad \ \mbox{ on }\partialΩ. \end{cases} \end{eqnarray*} By using a critical point theorem due to Ding in [Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal, 25(11)(1995)1095-1113], we obtain that system has infinitely many solutions under the sub-$(p,q)$ conditions.

math.AP↗
arXiv2025

Infinitely many solutions for a biharmonic-Kirchhoff system on locally finite graphs

The study on the partial differential equations (systems) in the graph setting is a hot topic in recent years because of their applications to image processing and data clustering. Our motivation is to develop some existence results for biharmonic-Kirchhoff systems and biharmonic systems in the Euclidean setting, which are the continuous models, to the corresponding systems in the locally finite graph setting, which are the discrete models. We mainly focus on the existence of infinitely many solutions for a biharmonic-Kirchhoff system on a locally finite graph. The method is variational and the main tool is the symmetric mountain pass theorem. We obtain that the system has infinitely many solutions when the nonlinear term admits the super-$4$ linear growth, and we also present the corresponding results to the biharmonic system. We also find that the results in the locally finite graph setting are better than that in the Euclidean setting, which caused by the better embedding theorem in the locally finite graph.

math.AP↗
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↗
arXiv2025

Multiplicity and uniqueness of positive solutions for a superlinear-singular $(p,q)$-Laplacian equation on locally finite graphs

We investigate the multiplicity and uniqueness of positive solutions for the superlinear singular $(p,q)$-Laplacian equation \begin{eqnarray*} \begin{cases} -Δ_p u-Δ_q u+a(x)u^{p-1}+b(x)u^{q-1}=f(x)u^{-γ}+λg(x)u^α, \;\;\;\;\hfill \mbox{in}\;\; V,\\ u>0,\;\;u\in W_a^{1,p}(V) \cap W_b^{1,q}(V), \end{cases} \end{eqnarray*} on a weighted locally finite graph $G=(V,E)$, where $0<γ<1 0, g \geq 0$, $f\in L^1(V)\cap L^{\frac{p}{p-1+γ}}(V) \cap L^{\frac{q}{q-1+γ}}(V)$ and $g\in L^1(V)\cap L^\infty(V)$. By making use of the method of Nehari manifold and the Ekeland's variational principle, we prove that there exist two positive solutions for $λ$ belonging to some precise interval. Besides, we also investigate the existence and uniqueness of positive solution for $λ<0$. We overcome some difficulties which are caused by: $(i)$ the singular term; $(ii)$ the definition of gradient $|\nabla u|$ on graph which is different from that on $\mathbb{R}^N$; $(iii)$ the lack of compactness of Sobolev embedding.

math.AP↗
arXiv2025

Least energy solutions of two asymptotically cubic Kirchhoff equations on locally finite graphs

We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=λu+η|u|^2u$ on a locally weighted and connected finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F(u)}{u^4}\to +\infty$ as $|u|\to\infty$, where $F(u)=\int_0^uf(s)ds$, nor $\frac{f(u)}{u}\to 0$ as $u\to 0$. By utilizing the constrained variational method, we prove that there exist $λ_1\ge 0$ and $η_0\ge 0$ ($λ_1^*\ge 0$ and $η_0^*\ge 0$) such that these two equations have at least a least energy solution if $|λ| η_0$ ($η>η_0^*$).

math.AP↗
arXiv2025

Multiplicity result on a class of nonhomogeneous quasilinear elliptic system with small perturbations in $\mathbb{R}^N$

We investigate a class of quasilinear elliptic system involving a nonhomogeneous differential operator which is introduced by C. A. Stuart [Milan J. Math. 79 (2011), 327-341] and depends on not only $\nabla u$ but also $u$. We show that the existence of multiple small solutions when the nonlinear term $F(x,u,v)$ satisfies locally sublinear and symmetric conditions and the perturbation is any continuous function with a small coefficient and no any growth hypothesis. Our technical approach is mainly based on a variant of Clark's theorem without the global symmetric condition. We develop the Moser's iteration technique to this quasi-linear elliptic system with nonhomogeneous differential operators and obtain that the relationship between $\|u\|_{\infty}$, $\|v\|_{\infty}$ and $\|u\|_{2^{\ast}}$, $\|v\|_{2^{\ast}}$. We overcome some difficulties which are caused by the nonhomogeneity of the differential operator and the lack of compactness of the Sobolev embedding.

math.AP↗
arXiv2024

Nontrivial solutions for a generalized poly-Laplacian system on finite graphs

We investigate the existence and multiplicity of solutions for a class of generalized coupled system involving poly-Laplacian and a parameter $λ$ on finite graphs. By using mountain pass lemma together with cut-off technique, we obtain that system has at least a nontrivial weak solution $(u_λ,v_λ)$ for every large parameter $λ$ when the nonlinear term $F(x,u,v)$ satisfies superlinear growth conditions only in a neighborhood of origin point $(0,0)$. We also obtain a concrete form for the lower bound of parameter $λ$ and the trend of $(u_λ,v_λ)$ with the change of parameter $λ$. Moreover, by using a revised Clark's theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every $λ>0$ when the nonlinear term $F(x,u,v)$ satisfies sublinear growth conditions only in a neighborhood of origin point $(0,0)$.

math.AP↗
arXiv2024

Nontrivial solutions for a $(p,q)$-Kirchhoff type system with concave-convex nonlinearities on locally finite graphs

By using the well-known mountain pass theorem and Ekeland's variational principle, we prove that there exist at least two fully-non-trivial solutions for a $(p,q)$-Kirchhoff elliptic system with the Dirichlet boundary conditions and perturbation terms on a locally weighted and connected finite graph $G=(V,E)$.We also present a necessary condition of the existence of semi-trivial solutions for the system. Moreover, by using Ekeland's variational principle and Clark's Theorem, respectively, we prove that the system has at least one or multiple semi-trivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions.

math.AP↗
arXiv2024

Multiple solutions for a class of nonhomogeneous elliptic systems with Dirichlet boundary or Neumann boundary

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain $Ω\subset\mathbb{R}^N $ and $N\geq 1$. We exploit the method which is based on [6]. This method let us obtain the concrete open interval about the parameter $λ$. Since the quasilinear term depends on $u$ and $\nabla u$, it is necessary for our proofs to use the theory of monotone operators and the skill of adding one dimension to space.

math.AP↗
arXiv2024

Dependence on parameters of solutions for a generalized poly-Laplacian system on weighted graphs

We mainly investigate the continuous dependence on parameters of nontrivial solutions for a generalized poly-Laplacian system on the weighted finite graph $G=(V, E)$. We firstly present an existence result of mountain pass type nontrivial solutions when the nonlinear term $F$ satisfies the super-$(p, q)$ linear growth condition which is a simple generalization of those results in [28]. Then we mainly show that the mountain pass type nontrivial solutions of the poly-Laplacian system are uniformly bounded for parameters and the concrete upper and lower bounds are given, and are continuously dependent on parameters. Similarly, we also present the existence result, the concrete upper and lower bounds, uniqueness, and dependence on parameters for the locally minimum type nontrivial solutions. Subsequently, we present an example on optimal control as an application of our results. Finally, we give a nonexistence result and some results for the corresponding scalar equation.

math.AP↗
arXiv2024

Infinitely many solutions for two generalized poly-Laplacian systems on weighted graphs

We investigate the multiplicity of solutions for a generalized poly-Laplacian system on weighted finite graphs and a generalized poly-Laplacian system with Dirichlet boundary value on weighted locally finite graphs, respectively, via the variational methods which are based on mountain pass theorem and topological degree theory. We obtain that these two systems have a sequence of minimax type solutions $\{(u_n,v_n)\}$ satisfying the energy functional $φ(u_n,v_n)\to +\infty$ as $n\to +\infty$ and a sequence of local minimum type solutions $\{(u_m^*,v_m^*)\}$ satisfying the energy functional $φ(u_m^*,v_m^*)\to -\infty$ as $m\to +\infty$.

math.AP↗
arXiv2024

Existence and convergence of ground state solutions for a $(p,q)$-Laplacian system on weighted graphs

We investigate the existence of ground state solutions for a $(p,q)$-Laplacian system with $p,q>1$ and potential wells on a weighted locally finite graph $G=(V,E)$. By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term $F$ takes on the super-$(p, q)$-linear growth and the potential functions $a(x)$ and $b(x)$ satisfy some suitable conditions, then for any fixed parameter $λ\geq1$, the system is provided with a ground state solution $(u_λ, v_λ)$. Additionally, we set up the convergence property of the solutions set $\{(u_λ, v_λ)\}$ when $λ\rightarrow +\infty$.

math.AP↗
arXiv2024

Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs

We investigate a generalized poly-Laplacian system with a parameter on weighted finite graph, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a $(p,q)$-Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 2010, 52(1-2): 152-160], which is an abstract critical points theorem without compactness condition, to obtain that these three systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.

math.AP↗
arXiv2024

Ground state sign-changing solutions for Kirchhoff-type equations with logarithmic nonlinearity on locally finite graphs

We obtain the existence results of ground state sign-changing solutions and ground state solutions for a class of Kirchhoff-type equations with logarithmic nonlinearity on a locally finite graph $G=(V,E)$, and obtain the sign-changing ground state energy is larger than twice of the ground state energy. The method we used is a direct non-Nehari manifold method in [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.]

math.AP↗
arXiv2023

Two nontrivial solutions for a nonhomogeneous quasilinear elliptic system with sign-changing weight functions

We are interested in looking for two nontrivial solutions for a class of nonhomogeneous quasilinear elliptic system with sign-changing weight functions and concave-convex nonlinearities on the bounded domain. This kind of quasilinear elliptic system arises from nonlinear optics, whose feature is that its differential operator depends on not only $\nabla u$ but also $u$. Employing the mountain pass theorem and Ekeland's variational principle as the major tools, we show that the system has at least one nontrivial solution of positive energy and one nontrivial solution of negative energy, respectively.

math.AP↗