A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

Advances in Nonlinear Analysis - Tập 8 Số 1 - Trang 645-660 - 2017

Alessio Fiscella¹

¹Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda 651, Campinas, SP CEP 13083-859 Brazil

Tóm tắt

AbstractIn 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)}

andMmodels 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 functionMis zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

Từ khóa

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