Multiplicity of k-convex solutions for a singular k-Hessian system

Abstract

In this article, we study the following nonlinear kk-Hessian system with singular weights Sk1k(σ(D2u1))=λb(∣x∣)f(−u1,−u2),inΩ,Sk1k(σ(D2u2))=λh(∣x∣)g(−u1,−u2),inΩ,u1=u2=0,on∂Ω,\left\{\begin{array}{ll}{S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{1}))=\lambda b\left(| x| )f\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{2}))=\lambda h\left(| x| )g\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {u}_{1}={u}_{2}=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where λ>0\lambda \gt 0, 1≤k≤N1\le k\le N is an integer, Ω\Omega stands for the open unit ball in RN{{\mathbb{R}}}^{N}, and Sk(σ(D2u)){S}_{k}(\sigma ({D}^{2}u)) is the kk-Hessian operator of uu. By using the fixed point index theory, we prove the existence and nonexistence of negative kk-convex radial solutions. Furthermore, we establish the multiplicity result of negative kk-convex radial solutions based on a priori estimate achieved. More precisely, there exists a constant λ∗>0{\lambda }^{\ast }\gt 0 such that the system admits at least two negative kk-convex radial solutions for λ∈(0,λ∗)\lambda \in \left(0,{\lambda }^{\ast }).

Keywords

• k-hessian system
• k-convex solution
• multiplicity
• upper and lower solution
• 35j60
• 35j75
• 35b09
• 35a16