arXiv subjects

Dong-Lun Wu

Publications and source records attributed to Dong-Lun Wu.

3 recordsLinked to original sources

arXiv2020

Multiple solutions for superlinear Klein-Gordon-Maxwell equations

In this paper, we consider the following Klein-Gordon-Maxwell equations \begin{eqnarray*} \left\{ \begin{array}{ll} -Δu+ V(x)u-(2ω+ϕ)ϕu=f(x,u)+h(x)&\mbox{in $\mathbb{R}^{3}$},\\ -Δϕ+ ϕu^2=-ωu^2&\mbox{in $\mathbb{R}^{3}$}, \end{array} \right. \end{eqnarray*} where $ω>0$ is a constant, $u$, $ϕ: \mathbb{R}^{3}\rightarrow \mathbb{R}$, $V : \mathbb{R}^{3} \rightarrow\mathbb{R}$ is a potential function. By assuming the coercive condition on $V$ and some new superlinear conditions on $f$, we obtain two nontrivial solutions when $h$ is nonzero and infinitely many solutions when $f$ is odd in $u$ and $h\equiv0$ for above equations.

math.DS↗

arXiv2019

Solutions for fourth-order Kirchhoff type elliptic equations involving concave-convex nonlinearities in $\mathbb{R}^{N}$

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} Δ^{2}u-M(\|\nabla u\|_{2}^{2})Δu+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N}, \end{eqnarray*} where $M(t):\mathbb{R}\rightarrow\mathbb{R}$ is the Kirchhoff function, $f(x,u)=λk(x,u)+ h(x,u)$, $λ\geq0$, $k(x,u)$ is of sublinear growth and $h(x,u)$ satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for $λ=0$. For $λ>0$ small enough, we obtain at least two nontrivial solutions. Furthermore, if $f(x,u)$ is odd in $u$, we show that above equations possess infinitely many solutions for all $λ\geq0$. Our theorems generalize some known results in the literatures even for $λ=0$ and our proof is based on the variational methods.

math.DS↗

arXiv2019

New homoclinic orbits for Hamiltonian systems with asymptotically quadratic growth at infinity

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

math.DS↗