Liouville type theorem for a singular elliptic equation with finite Morse index

Abstract

Abstract This paper considers the nonexistence of solutions for the following singular quasilinear elliptic problem: 0.1 {−div(|x|−ap|∇u|p−2∇u)=f(|x|)|u|r−1u,x∈R+N,|x|−ap|∇u|p−2∂u∂ν=g(|x|)|u|q−1u,on ∂R+N, $$\begin{aligned} \textstyle\begin{cases} -\operatorname{div} ( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= f( \vert x \vert ) \vert u \vert ^{r-1}u, \quad x\in {\mathbb {R}} ^{N}_{+}, \\ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\frac{\partial u}{\partial \nu }=g( \vert x \vert ) \vert u \vert ^{q-1}u, \quad \text{on } \partial {\mathbb {R}} ^{N}_{+}, \end{cases}\displaystyle \end{aligned}$$ where R+N={x=(x′,xN)|x′∈RN−1,xN>0} ${\mathbb {R}} ^{N}_{+}=\{x=(x',x_{N})| x'\in {\mathbb {R}} ^{N-1}, x_{N}>0 \}$ and ∂R+N={x=(x′,xN)|x′∈RN−1,xN=0} $\partial {\mathbb {R}} ^{N}_{+}=\{x=(x',x_{N})| x'\in {\mathbb {R}} ^{N-1}, x_{N}=0\}$. When the weight functions satisfy some suitable assumptions, we prove that problem (0.1) has no nontrivial bounded solutions with finite Morse index.

Keywords

• Liouville theorem
• Morse index
• Singular elliptic equation