Fading absorption in non-linear elliptic equations

Abstract

We study the equation $-\Delta u+h(x)|u{|}^{q-1}u=0$, $q>1$, in $R^N_+=R^{N-1}\ti R_+$ where $h\in C(\overline{R_{+}^N})$, $h\ge 0$. Let $(x_1,...,x_N)$ be a coordinate system such that $R_{+}^N=[x_N>0]$ and denote a point $x\in \RN$ by $(x^{\prime },x_N)$. Assume that $h(x^{\prime },x_N)>0$ when $x^{\prime }\ne 0$ but $h(x^{\prime },x_N)\to 0$ as $|x^{\prime }|\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.

Cited by in corpus (1)

• Generalization of the Keller-Osserman theorem for higher order differential inequalities