Best constant of 3D Anisotropic Sobolev inequality and its applications
Jianqing Chen
In this paper, we firstly determine the best constant of the three dimensional anisotropic Sobolev inequality [2]; then we use this best constant to investigate qualitative conditions for the uniform bound of the solution of the generalized Kadomtsev-Petviashvili (KP) I equation in three dimensions. The (KP) I equation is a model for the propagation of weakly nonlinear dispersive long waves on the surface of a fluid, when the wave motion is essentially one- directional and weak transverse effects are taken into account [11, 10]. Our results improve and optimize previous works [6, 12, 13, 14, 15].
keywords: uniform bound of the solution. Best constant of the anisotropic Sobolev inequality Kadomtsev- Petviashvili equation
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations
Jianqing Chen Boling Guo
In this paper, we first give an important interpolation inequality. Secondly, we use this inequality to prove the existence of local and global solutions of an inhomogeneous Schrödinger equation. Thirdly, we construct several invariant sets and prove the existence of blowing up solutions. Finally, we prove that for any $\omega>0$ the standing wave $e^{i \omega t} \phi (x)$ related to the ground state solution $\phi$ is strongly unstable.
keywords: global existence strong instability. blowing up Interpolation inequality
A variational argument to finding global solutions of a quasilinear Schrödinger equation
Jianqing Chen
We prove a sharp existence result of global solutions of the quasilinear Schrödinger equation

$iu_t + u_{x x} + |u|^{p-2}u +(|u|^2)_{x x}u = 0,\quad u|_{t=0}=u_0(x),\quad x\in \mathbb R$

for a large class of initial data. The result gives a qualitative description on how small an initial data can ensure the existence of global solutions which sharpen a global existence result with small initial data [7, 10].

keywords: Global solutions quasilinear Schrödinger equation.
Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential
Yaoping Chen Jianqing Chen
In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form:
$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p,\ \ x∈Ω\backslash\{0\}$
with Dirichlet boundary conditions, where
$0∈ Ω\subset\mathbb{R}^N $
$N≥q 3 $
) be a bounded domain with smooth boundary
$\partial Ω $
$μ>0 $
is a parameter,
$0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $
$h(x)>0 $
$W(x) $
is a given function with the set
$\{x∈ Ω: W(x)>0\} $
of positive measure.
keywords: Elliptic problems Hardy term multiple positive solutions extremal values uniform estimates
Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity
Jianqing Chen
In this paper, we first give a sharp variational characterization to the smallest positive constant $C_{VGN}$ in the following Variant Gagliardo-Nirenberg interpolation inequality: $$ \int_{\mathbb{R}^N\times\mathbb{R}^N}{{|u(x)|^p|u(y)|^p}\over{|x-y|^\alpha}}dxdy\leq C_{VGN} \|\nabla u\|_{L^2}^{N(p-2)+\alpha} \|u\|_{L^2}^{2p-(N(p-2)+\alpha)}, $$ where $u\in W^{1,2}(\mathbb{R}^N)$ and $N\geq 1$. Then we use this characterization to determine the sharp threshold of $\|\varphi_0\|_{L^2}$ such that the solution of $i\varphi_t = - \triangle \varphi + |x|^2\varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$ with initial condition $\varphi(0, x) = \varphi_0$ exists globally or blows up in a finite time. We also outline some results on the applications of $C_{VGN}$ to the Cauchy problem of $i\varphi_t = - \triangle \varphi - \varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.
keywords: Variant Gagliardo-Nirenberg interpolation inequality sharp variational characterization minimal action solution Hartree type nonlinearity.

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