Isolated Singularities of Polyharmonic Inequalities

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.Comment: 18 page

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity $$\Delta^m u = \pm u^{\alpha} \quad \text{ in } \mathbb R^n$$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $\alpha$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $\alpha$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.Comment: 18 pages, 0 figur

Isolated Singularities of Polyharmonic Operator in Even Dimension

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