Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth,Advanced Nonlinear Studies

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Existence of Solutions to Fractional p-Laplacian Systems with Homogeneous Nonlinearities of Critical Sobolev Growth
Advanced Nonlinear Studies ( IF 2.1 ) Pub Date : 2020-08-01 , DOI: 10.1515/ans-2020-2098
Guozhen Lu ₁ , Yansheng Shen ₂

Affiliation

Abstract In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { ( - Δ p ) s ⁢ u = Q u ⁢ ( u , v ) + H u ⁢ ( u , v ) in ⁢ Ω , ( - Δ p ) s ⁢ v = Q v ⁢ ( u , v ) + H v ⁢ ( u , v ) in ⁢ Ω , u = v = 0 in ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where ( - Δ p ) s {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, p > 1 {p>1} , s ∈ ( 0 , 1 ) {s\in(0,1)} , p ⁢ s < N {ps

中文翻译：

具有临界 Sobolev 增长的齐次非线性的分数 p-Laplacian 系统解的存在性

\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. 其中 ( - Δ p ) s {(-\Delta_{p})^{s}} 表示分数 p-Laplacian 算子，p > 1 {p>1} , s ∈ ( 0 , 1 ) {s\in( 0,1)} , p ⁢ s < N {ps

更新日期：2020-08-01

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