On the existence and multiplicity of solutions to fractional Lane-Emden elliptic systems involving measures

Advances in Nonlinear Analysis

article

Mousomi Bhakta¹  Phuoc-Tai Nguyen² 

[1] Department of Mathematics, Indian Institute of Science Education and Research;Department of Mathematics and Statistics, Masaryk University

关键词: nonlocal;    system;    existence;    multiplicity;    linking theorem;    measure data;    source terms;    positive solution;   

DOI  :  10.1515/anona-2020-0060

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We study positive solutions to the fractional Lane-Emden system (− Δ )su=vp+μ in Ω (− Δ )sv=uq+ν in Ω u=v=0in Ω c=RN∖ Ω , $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C 2 bounded domains in ℝ N , s ∈ (0, 1), N > 2 s , p > 0, q > 0 and μ , ν are positive measures in Ω . We prove the existence of the minimal positive solution of (S) under a smallness condition on the total mass of μ and ν . Furthermore, if p , q ∈ (1,N+sN− s) $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ , ν ∈ L r ( Ω ), for some r > N2s, $\begin{array}{} \frac{N}{2s}, \end{array}$ we show the existence of at least two positive solutions of (S). The novelty lies at the construction of the second solution, which is based on a highly nontrivial adaptation of Linking theorem. We also discuss the regularity of the solutions.

【 授权许可】

CC BY