Isolated Singularities of Polyharmonic Operator in Even Dimension

Abstract

We consider the equation $\Delta^2 u=g(x,u) \geq 0$ in the sense of distribution in $\Omega'=\Omega\setminus \{0\} $ where $u$ and $ -\Delta u\geq 0.$ Then it is known that $u$ solves $\Delta^2 u=g(x,u)+\alpha \delta_0-\beta \Delta \delta_0,$ for some non-negative constants $\alpha$ and $ \beta.$ In this paper we study the existence of singular solutions to $\Delta^2 u= a(x) f(u)+\alpha \delta_0-\beta \Delta \delta_0$ in a domain $\Omega\subset \mathbb{R}^4,$ $ a$ is a non-negative measurable function in some Lebesgue space. If $\Delta^2 u=a(x)f(u)$ in $\Omega',$ then we find the growth of the nonlinearity $f$ that determines $\alpha$ and $\beta$ to be $0.$ In case when $\alpha=\beta =0,$ we will establish regularity results when $f(t)\leq C e^{\gamma t},$ for some $C, \gamma>0.$ 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 $a(x)=|x|^\sigma.$ Later we discuss its analogous generalization for the polyharmonic operator.