Isolated Singularities of Polyharmonic Operator in Even Dimension

We consider the equation $Δ^2 u=g(x,u) \geq 0$ in the sense of distribution in $Ω'=Ω\setminus \{0\}$ where $u$ and $-Δu\geq 0.$ Then it is known that $u$ solves $Δ^2 u=g(x,u)+αδ_0-βΔδ_0,$ for some non-negative constants $α$ and $β.$ In this paper we study the existence of singular solutions to $Δ^2 u= a(x) f(u)+αδ_0-βΔδ_0$ in a domain $Ω\subset \mathbb{R}^4,$ $a$ is a non-negative measurable function in some Lebesgue space. If $Δ^2 u=a(x)f(u)$ in $Ω',$ then we find the growth of the nonlinearity $f$ that determines $α$ and $β$ to be $0.$ In case when $α=β=0,$ we will establish regularity results when $f(t)\leq C e^{γt},$ for some $C, γ>0.$ This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions $(N\geq 5)$ with a specific weight function $a(x)=|x|^σ.$ Later we discuss its analogous generalization for the polyharmonic operator.

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