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

Advances in Nonlinear Analysis

article

Alessio Fiscella¹ 

[1] Universidade Estadual de Campinas

关键词: Kirchhoff-type problems;    fractional Laplacian;    singularities;    critical nonlinearities;    perturbation methods;   

DOI  :  10.1515/anona-2017-0075

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In this paper, we consider the following critical nonlocal problem: { M ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ ? x ⁢ ? y ) ⁢ ( - Δ ) s ⁢ u = λ u γ + u 2 s * - 1 in  ⁢ Ω , u > 0 in  ⁢ Ω , u = 0 in  ⁢ ℝ N ∖ Ω , \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 ℝ N {\mathbb{R}^{N}} with continuous boundary, dimension N > 2 ⁢ s {N>2s} with parameter s ∈ ( 0 , 1 ) {s\in(0,1)} , 2 s * = 2 ⁢ N / ( N - 2 ⁢ s ) {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, λ > 0 {\lambda>0} is a real parameter, γ ∈ ( 0 , 1 ) {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while ( - Δ ) s {(-\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.

【 授权许可】

CC BY