Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent

Abstract

Abstract In this paper, we study the following critical system with fractional Laplacian: {(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω, $$\textstyle\begin{cases} (-\Delta)^{s}u+\lambda_{1}u=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha \gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} & \text{in } \Omega, \\ (-\Delta)^{s}v+\lambda_{2}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta \gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v & \text{in } \Omega, \\ u=v=0 & \text{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} $$ where (−Δ)s $(-\Delta)^{s}$ is the fractional Laplacian, 00 $\mu_{1},\mu_{2}>0$, 2∗=2NN−2s $2^{\ast}=\frac{2N}{N-2s}$ is a fractional critical Sobolev exponent, N>2s $N>2s$, 1−λ1,s(Ω) $\lambda_{1},\lambda_{2}>-\lambda_{1,s}(\Omega)$, λ1,s(Ω) $\lambda_{1,s}(\Omega)$ is the first eigenvalue of the non-local operator (−Δ)s $(-\Delta)^{s}$ with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all γ>0 $\gamma>0$. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when γ→0 $\gamma\rightarrow0$.

Keywords

• Fractional Laplacian
• Critical exponent
• Ground state solution
• Higher energy solution