The role of the power 3 for elliptic equations with negative exponents

Abstract

Let \(\Omega \subset {\mathbb{R }}^{N}\) be a bounded regular domain of dimension \(N\ge 3,\;h\) a positive \(L^{1}\) function on \(\Omega .\) Elliptic equations of singular growth like $$\begin{aligned} -\Delta u=\frac{h(x)}{u^{p}}\quad \mathrm{in}\;\Omega , \quad u>0\quad \mathrm{in}\;\Omega , \quad u=0\quad \mathrm{on}\;\partial \Omega , \end{aligned}$$ have been the target of investigation for decades. A very nice result for existence of solutions of such an equation is due to Lazer–McKenna (Proc AMS 111:720–730, 1991) when \(h\) is a positive continuous function on \(\overline{\Omega }.\) In that paper the Lazer–McKenna obstruction was first presented: the equation has a \(H_{0}^{1}\)-solution if and only if \(p<3.\) In this paper we provide an extension of the classical Lazer–McKenna obstruction and reveal the role of 3. Moreover, we give a local description of the solution set.