Affiliation:
1. Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP CEP 13083-859, Brazil
Abstract
Abstract
In this paper, we consider the following critical nonlocal problem:
\left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{%
\lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)%
^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text%
{in }\Omega,\\
\displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\
\displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{%
N}\setminus\Omega,\end{aligned}\right.
where Ω is an open bounded subset of
{\mathbb{R}^{N}}
with continuous boundary, dimension
{N>2s}
with parameter
{s\in(0,1)}
,
{2^{*}_{s}=2N/(N-2s)}
is the fractional critical Sobolev exponent,
{\lambda>0}
is a real parameter,
{\gamma\in(0,1)}
and M models a Kirchhoff-type coefficient, while
{(-\Delta)^{s}}
is the fractional Laplace operator.
In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.
Funder
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Reference54 articles.
1. Semilinear problems for the fractional Laplacian with a singular nonlinearity;Open Math.,2015
2. Multiple positive solutions for Kirchhoff type problems with singularity;Commun. Pure Appl. Anal.,2013
3. Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator;Differential Integral Equations,2016
4. The Brezis–Nirenberg result for the fractional Laplacian;Trans. Amer. Math. Soc.,2015
Cited by
68 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献