arXiv subjects
Publications and source records attributed to Yongsheng Jiang.
This paper is motivated by a gauged Schrödinger equation in dimension 2. We are concerned with radial stationary states under the presence of a vortex at the origin. Those states solve a nonlinear nonlocal PDE with a variational structure. We will study the global behavior of that functional, extending known results for the regular case.
The paper considers the following nonhomogeneous Schrödinger-Maxwell system -Δu + u+λϕ(x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -Δϕ= u^2, \ x\in \mathbb{R}^3, . \leqno{(SM)} where $λ>0$, $p\in(1,5)$ and $g(x)=g(|x|)\in L^2(\mathbb{R}^3)\setminus{0}$. There seems no any results on the existence of multiple solutions to problem (SM) for $p \in (1,3]$. In this paper, we find that there is a constant$C_p>0$ such that problem (SM) has at least two solutions for all $p\in (1,5)$ provided $\|g\|_{L^2} \leq C_p$, but only for $p\in(1,2]$ we need $λ>0$ is small. Moreover, $C_p=\frac{(p-1)}{2p}[\frac{(p+1)S^{p+1}}{2p}]^{1/(p-1)}$, where $S$ is the Sobolev constant.
The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schrödinger-Poisson system {ll} -Δu +(λ+\frac{1}{|y|^α})u+ϕ(x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -Δϕ= u^2,\ \lim\limits_{|x|\rightarrow +\infty}ϕ(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $λ\geqslant0$, $α\in[0,8)$ and $\max\{2,\frac{2+α}{2}\}<p<5$.
We study the following Schrödinger-Poisson system (P_λ)\{ll} -Δu + (1+μg(x))u+λϕ(x) u =|u|^{p-1}u, x\in \mathbb{R}^3,
We study the following Schrödinger-Poisson system (P_λ){ll} -Δu + V(x)u+λϕ(x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -Δϕ= u^2, \lim\limits_{|x|\to +\infty}ϕ(x)=0, u>0, where $λ\geqslant0$ is a parameter, $1 < p < +\infty$, $V(x)$ and $Q(x)$ are sign-changing or non-positive functions in $ L^{\infty}(\mathbb{R}^3)$. When $V(x)\equiv Q(x)\equiv1$, D.Ruiz \cite{RuizD-JFA} proved that ($P_λ$) with $p\in(2,5)$ has always a positive radial solution, but ($P_λ$) with $p\in(1,2]$ has solution only if $λ>0$ small enough and no any nontrivial solution if $λ\geqslant{1/4}$. By using sub-supersolution method, we prove that there exists $λ_0>0$ such that ($P_λ$) with $p\in(1,+\infty)$ has always a bound state ($H^1(\mathbb{R}^3)$ solution) for $λ\in[0,λ_0)$ and certain functions $V(x)$ and $Q(x)$ in $ L^{\infty}(\mathbb{R}^3)$. Moreover, for every $λ\in[0,λ_0)$, the solutions $u_λ$ of $\rm (P_λ)$ converges, along a subsequence, to a solution of ($P_0$) in $H^1$ as $λ\to 0$.