## Journals

- Advances in Mathematics of Communications
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CPAA

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].

DCDS-B

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.

CPAA

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].

CPAA

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

(

) be a bounded domain with smooth boundary

,

is a parameter,

,

and

is a given function with the set

of positive measure.

$0∈ Ω\subset\mathbb{R}^N $ |

$N≥q 3 $ |

$\partial Ω $ |

$μ>0 $ |

$0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $ |

$h(x)>0 $ |

$W(x) $ |

$\{x∈ Ω: W(x)>0\} $ |

keywords:
Elliptic problems
,
Hardy term
,
multiple positive solutions
,
extremal values
,
uniform estimates

DCDS-S

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
$i\varphi_t = - \triangle \varphi -
\varphi|\varphi|^{p-2}(|x|^{-\alpha}*|\varphi|^p)$.

## Year of publication

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