arXiv subjects

Juntao Sun

Publications and source records attributed to Juntao Sun.

21 records · Page 2Linked to original sources

arXiv2015

On the non-autonomous Schrödinger-Poisson problems in $\mathbb{R}^{3}$

In this paper, we study the problem: \begin{equation*} \left\{ \begin{array}{ll} -Δu+u+λK\left( x\right) ϕu=a\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -Δϕ=K\left( x\right) u^{2} & \ \text{in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $λ>0$ and $2 0$, we establish some existence results of positive solutions, depending on the parameter $λ$. More importantly, we prove the existence of ground state solutions for the case $3.18\thickapprox \frac{1+\sqrt{73}}{3}<p<4.$

math.AP↗

arXiv2014

On the Schrödinger-Poisson system with steep potential well and indefinite potential

In this paper, we study the following Schrödinger-Poisson system: $$ \left\{\aligned&-Δu+V_λ(x)u+K(x)ϕu=f(x,u)&\quad\text{in }\bbr^3,\\ &-Δϕ=K(x)u^2&\quad\text{in }\bbr^3,\\ &(u,ϕ)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_λ)} $$ where $V_λ(x)=λa(x)+b(x)$ with a positive parameter $λ$, $K(x)\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity $|u|^{p-2}u$. By applying the method of penalized functions, the existence of one nontrivial solution for such system in the less-studied case $3<p\leq4$ is obtained for $λ$ sufficiently large. The concentration behavior of this nontrivial solution for $λ\to+\infty$ are also observed. It is worth to point out that some new conditions on the potentials are introduced to obtain this nontrivial solution and the Schrödinger operator $-Δ+V_λ(x)$ may be strong indefinite in this paper.

math.AP↗

arXiv2014

On the indefinite Kirchhoff type problems with local sublinearity and linearity

The purpose of this paper is to study the indefinite Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ll} M\left( \int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -Δu+u\right] =f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left( \mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where $N\geq 1$, $M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $ f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ is \textquotedblleft local\textquotedblright\ sublinear at the origin and \textquotedblleft local\textquotedblright\ linear at infinite. Using the mountain pass theorem and Ekeland variational principle, the existence and multiplicity of nontrivial solutions are obtained. In particular, the criterion of existence of three nontrivial solutions is established.

math.AP↗