Geometrically distinct solutions for Klein–Gordon–Maxwell systems with super-linear nonlinearities

This paper is concerned with the following Klein–Gordon–Maxwell system $\left\{\begin{array}{c}-△u+V\left(x\right)u-(2\omega +\varphi )\varphi u=f(x,u),x\in {\mathbb{R}}^3,\hfill \\ △\varphi =(\omega +\varphi )u^2,x\in {\mathbb{R}}^3,\hfill \end{array}\right.$where $\omega >0$ is a constant, $V$ and $f$ are periodic with respect to $x$. By combining deformation type arguments, Lusternik–Schnirelmann theory and some new tricks, we prove that the above system admits infinitely many geometrically distinct solutions under weaker superlinear conditions instead of the common super-cubic conditions on $f$. Our result seems new and extends the previous results in the literature.     

Ground state solutions for Klein–Gordon–Maxwell system with steep potential well

In this paper, we study the following Klein–Gordon–Maxwell system $\left\{\begin{array}{c}-\Delta u+(\lambda a\left(x\right)+1)u-(2\omega +\varphi )\varphi u=f(x,u),\mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\mathrm{in}{\mathbb{R}}^3.\hfill \end{array}\right.$Using variational methods, we obtain the existence of ground state solutions under some appropriate assumptions on $a$ and $f$.     

Symmetric results of a Hénon type elliptic equation

In this paper, we study the following weighted elliptic system $\left\{\begin{array}{cc}\hfill & -\Delta u+u={\lambda }_1{\left|x\right|}^{\alpha }{u}^3+\mu {\left|x\right|}^{\beta }u{v}^2,x\in B,\hfill \\ \hfill & -\Delta v+v={\lambda }_2{\left|x\right|}^{\alpha }{v}^3+\mu {\left|x\right|}^{\beta }{u}^{2}v,x\in B,\hfill \\ \hfill & u,v>0,x\in B,u=v=0,x\in \partial B,\hfill \end{array}\right.$where $B\subset {\mathbb{R}}^N(N=2,3)$ is the unit ball centered at the origin, ${\lambda }_1,{\lambda }_2>0$, $\mu >0$, $\beta >0$, $\alpha >0$. By virtue of variational approaches and rescaling methods, the system has a nontrivial ground state solution with $\alpha >\beta >0$, moreover, by reduction methods, the ground state solution is radial symmetry if $\beta >0$ small enough.     

Asymptotic behavior of ground state solutions for nonlinear Schrödinger systems

Consider the nonlinear Schrödinger system $\left\{\begin{array}{cc}\hfill & -\Delta u_1+V_1\left(x\right)u_1={\mu }_1\left(x\right)u_1^3+\beta \left(x\right)u_1{u}_2^2\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & -\Delta u_2+V_2\left(x\right)u_2=\beta \left(x\right)u_1^2{u}_2+{\mu }_2\left(x\right)u_2^3\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & u_j\in H^1\left({\mathbb{R}}^N\right),j=1,2,\hfill \end{array}\right.$where $N=1,2,3$, and the potentials $V_j,{\mu }_j,\beta$ are periodic or $V_j$ are well-shaped and ${\mu }_j,\beta$ are anti-well-shaped. When the coupling coefficient $\beta$ is either small or large in terms of $V_j$ and ${\mu }_j$, existence of a positive ground state solution was proved in Liu and Liu (2015). In this paper, we describe the asymptotic behavior of ground state solutions when ${\left|\beta \right|}_{L^{\infty }\left({\mathbb{R}}^N\right)}$ tends to zero or ${min}_{{\mathbb{R}}^N}\beta$ tends to $+\infty$.     

Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption

In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source $\left\{\begin{array}{cc}{u}_t=\Delta \left(u\gamma \left(v\right)\right)+\rho u-\mu u^l,\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-vw,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=-\delta w+u,\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$, where the motility function $\gamma \left(v\right)\in C^3\left((0,+\infty )\right),\gamma \left(v\right)>0,{\gamma }^{\prime }\left(v\right)<0$ on $(0,+\infty )$, ${lim}_{v\to \infty }\gamma \left(v\right)=0$, $\rho >0,\mu >0,l>1$ and $\delta >0$. The purpose of this paper is to prove that if $l>max\{1,\frac{n}{2}\}$, then the system possesses a global solution. In addition, if $l$ satisfies $l\left\{\begin{array}{cc}\ge 2,\hfill & \text{if }n\le 3,\hfill \\ >\frac{n}{2},\hfill & \text{if }n\ge 4,\hfill \end{array}\right.$then the solution $(u,v,w)$ satisfies ${6u(\cdot ,t)-{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}+{6v(\cdot ,t)6}_{L^{\infty }\left(\Omega \right)}+{6w(\cdot ,t)-\frac{1}{\delta }{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}$$\to 0ast\to \infty .$     

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of

$\{\begin{array}{ccc}?\mathrm{\Delta }u(y)\hfill & \hfill =\hfill & u{(y)}^p\text{ in }y\in {\mathrm{\Omega }}_t,\hfill \\ u\hfill & \hfill =\hfill & 0\text{ on }y\in ?{\mathrm{\Omega }}_t,\hfill \end{array}$

where ${\mathrm{\Omega }}_t$ is a sufficiently small $C^{2,\alpha }$ perturbation of the cone $\mathrm{\Omega }:=\{x\in {\mathbb{R}}^N:x=r\theta ,r>0,\theta \in S\}$ where $S?S^{N?1}$ has a smooth nonempty boundary and where $p>1$ satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.