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

Abstract

Abstract We study positive solutions to the fractional Lane-Emden system $$\begin{array}{} \displaystyle \left\{ \begin{aligned} (-{\it\Delta})^s u &= v^p+\mu \quad &\text{in } {\it\Omega} \\(-{\it\Delta})^s v &= u^q+\nu \quad &\text{in } {\it\Omega}\\u = v &= 0 \quad &&\!\!\!\!\!\!\!\!\!\!\!\!\text{in } {\it\Omega}^c={\mathbb R}^N \setminus {\it\Omega}, \end{aligned} \right. \end{array}$$(S) where Ω is a C2 bounded domains in ℝN, s ∈ (0, 1), N > 2s, 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 ∈ $\begin{array}{} (1,\frac{N+s}{N-s}) \end{array}$ and 0 ≤ μ, ν ∈ Lr(Ω), for some r > $\begin{array}{} \frac{N}{2s}, \end{array}$ 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.