arXiv subjects

Jiabao Su

Publications and source records attributed to Jiabao Su.

6 recordsLinked to original sources

arXiv2026

Ground state of indefinite coupled nonlinear Schrödinger systems

In this paper, we study the ground state solutions of the following coupled nonlinear Schrödinger system (P) $-Δu_1-τ_1 u_1 =μ_1u_1^3+βu_1u_2^2$, $ -Δu_2-τ_2 u_2 =μ_2u_2^3+βu_1^2u_2$ in $Ω$, $u_1=u_2=0$ on $\partialΩ$, where $μ_1, μ_2>0$, $β>0$ and $Ω\subset \mathbb{R}^N (N\le3)$ is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., $τ_1, τ_2$ are greater than or equal to the principal eigenvalue of $-Δ$ with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to $(P)$, and also provide information on critical energy levels for coupling parameter $β$ in some ranges.

math.AP↗

arXiv2024

The existence of ground state solutions for critical Hénon equations in $\mathbb{R}^N$

In this paper we confirm that $2^*(γ)=\frac{2(N+γ)}{N-2}$ with $γ>0$ is exactly the critical exponent for the embedding from $H_r^1(\mathbb{R}^N)$ into $L^q(\mathbb{R}^N;|x|^γ)$($N\geqslant 3$) (see \cite{2007SWW-1,2007SWW-2}) and name it as the upper Hénon-Sobolev critical exponent. Based on this fact we study the ground state solutions of critical Hénon equations in $\mathbb{R}^N$ via the Nehari manifold methods and the great idea of Brezis-Nirenberg in \cite{1983BN}. We establish the existence of the positive radial ground state solutions for the problem with one single upper Hénon-Sobolev critical exponent. We also deal with the existence of the nonnegative radial ground state solutions for the problems with multiple critical exponents, including Hardy-Sobolev critical exponents or Sobolev critical exponents or the upper Hénon-Sobolev critical exponents.

math.AP↗

arXiv2024

On the double critical Maxwell equations

In this paper, we focus on (no)existence and asymptotic behavior of solutions for the double critical Maxwell equation involving with the Hardy, Hardy-Sobolev, Sobolev critical exponents. The existence and noexistence of solutions completely depend on the power exponents and coefficients of equation. On one hand, based on the concentration-compactness ideas, applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the critical Maxwell equation for three different scenarios. On the other hand, for the case $λ<0$ and $0\leq s_2<s_1<2$, which is a type open problem raised by Li and Lin. Draw support from a changed version of Caffarelli-Kohn-Nirenberg inequality, we find that there exists a constant $λ^*$ which is a negative number having explicit expression, such that the problem has no nontrivial solution as the coefficient $λ<λ^*$. Moreover, there exists a constant $λ^*<λ^{**}<0$ such that, as $λ^{**}<λ<0$, the equation has a nontrivial solution using truncation methods. Furthermore, we establish the asymptotic behavior of solutions of equation as coefficient converges to zero for the all cases above.

math.AP↗

arXiv2024

Ground states for the double weighted critical Kirchhoff equation on the unit ball in $\mathbb{R}^3$

This paper deals with the existence of ground states for degenerative ($a=0$) and non-degenerative ($a>0$) double weighted critical Kirchhoff equation \begin{eqnarray*} \left\{ \begin{array}{ll} \displaystyle-\left(a+b\int_B |\nabla u|^2dx\right)Δu=|x|^{α_1} |u|^{4+2α_1}u+μ|x|^{α_2} |u|^{4+2α_2}u+λh(|x|) f(u) &{\rm in}\ B,\\ u=0 &{\rm on}\ \partial B, \end{array} \right. \end{eqnarray*} where $B$ is a unit open ball in $\mathbb{R}^3$ with center $0$, $a\geq0, b>0, μ\in \mathbb{R}, λ>0, α_1>α_2>-2$, $4+2α_i=2^*(α_i)-2\ (i=1,2)$ with $2^*(α_i)=\frac{2(N+α_i)}{N-2} $ $(N=3)$ being Hardy-Sobolev ($-2<α_i<0$), Sobolev ($α_i=0$) or Hénon-Sobolev ($α_i>0$) critical exponent of the embedding $H_{0,r}^1(B)\hookrightarrow L^p(B;|x|^{α_i})$. Noting that the sign of $μ$ gives rise to a great effect on the existence of solutions. The methods rely on Nehari manifold and the mountain pass theorem.

math.AP↗

arXiv2023

Existence of Nontrivial Solutions for the Nonlinear Equation on Locally Finite Graphs

Suppose that $G=(V, E)$ be a locally finite and connected graph with symmetric weight and uniformly positive measure, where $V$ denotes the vertex set and $E$ denotes the edge set. We are concered with the following problem $$ \begin{cases}-Δu+h u=f(x, u), & \text { in } Ω, \\ u=0, & \text { on } \partial Ω,\end{cases} $$ on the graph, where $h: Ω\rightarrow \mathbb{R}$, $f: Ω\times \mathbb{R} \rightarrow \mathbb{R}$ and $u: Ω\rightarrow \mathbb{R}$. When $ f $ and $ h $ satisfies certain assumption conditions, we can ascertain the existence of one or two nontrivial solutions on the graph.

math.FA↗

arXiv2023

The quasilinear Schrödinger--Poisson system

This paper deals with the $(p,q)$--Schrödinger--Poisson system \begin{eqnarray*} \left \{\begin{array}{ll} \displaystyle -Δ_p u+|u|^{p-2}u+λϕ|u|^{s-2}u=|u|^{r-2}u,&\mathrm{in} \ \mathbb{R}^3,\\ \displaystyle -Δ_q ϕ= |u|^s, &\mathrm{in}\ \mathbb{R}^3,\\ \end{array} \right. \end{eqnarray*} where $1 0$ is a parameter. This quasilinear system is new and has never been considered in the literature. The uniqueness of solutions of the quasilinear Poisson equation is obtained via the Minty--Browder theorem. The variational framework of the quasilinear system is built and the nontrivial solutions of the system are obtained via the mountain pass theorem.

math.AP↗