arXiv subjects

Lingyu Jin

Publications and source records attributed to Lingyu Jin.

6 recordsLinked to original sources

arXiv2019

A Global Compact Result for a Fractional Elliptic Problem with Hardy term and critical non-linearity on the whole space

In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$ (-Δ)^α u-μ\frac{u}{|x|^{2α}}+a(x) u=|u|^{2^*-2}u+k(x)|u|^{q-2}u$$ $$ u\,\in\,H^α({\mathbb R}^N),$$ where $2 4α$, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to $(*)$, we obtain the existence of positive solutions for $(*)$ under certain assumptions on $a(x),k(x)$.

math.AP↗

arXiv2017

A Hopf's Lemma and the Boundary Regularity for the Fractional P-Laplacian

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.

math.AP↗

arXiv2017

Large time behavior of solutions for a Cauchy problem on nonlinear conservation laws with large initial data in the whole space

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, frequency decomposition, pseudo-differential operators, we obtain the global existence and the optimal decay estimate of $t$.

math.AP↗

arXiv2017

The Brezis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.

math.AP↗

arXiv2017

The global existence and time-decay for the solution of the fractional pseudo-parabolic equation

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space $R^n,n\geq 1$. Here, the fractional order $α$ is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for $0<α< 1$ and $α> 1$, respectively. We establish the global existence and time-decay rates for small-amplitude classical solutions to the Cauchy problem for $α>0$. In the case that $0<α< 1$ , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.

math.AP↗

arXiv2017

A Global Compact Result for a Fractional Elliptic Problem with Critical Sobolev-Hardy Nonlinearities on ${\mathbb R}^N$

In this paper, we are concerned with the following type of elliptic problems: $$ (-Δ)^α u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u, u\,\in\,H^α({\mathbb R}^N), $$ where $2<q< 2^*$, $0<α<1$, $0<s<2α$, $2^*_{s}=2(N-s)/(N-2α)$ is the critical Sobolev-Hardy exponent, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to the problem, we obtain the existence of positive solutions under certain assumptions on $a(x),k(x)$.

math.AP↗