Singular semilinear elliptic inequalities in the exterior of a compact set (Q2866506)

This paper deals with the (non)existence of \(C^2\)-positive solutions of the following semilinear elliptic inequality (1) NEWLINE\[NEWLINE-\Delta u\geq\varphi(\delta_K(x))f(u)\quad\text{in }\mathbb{R}^N\setminus K,\;N\geq 3,NEWLINE\]NEWLINE where \(\varphi\), \(f\) are positive and nonincreasing continuous functions, \(\delta_K(x)= \text{dist}(x,\partial K)\) and the compact set \(K\subset\mathbb{R}^N\) has finitely many connected components, each of which is either the closure of a \(C^2\)-domain or an isolated point. In the Theorems 1.1 and 1.4 the authors propose optimal conditions on \(\varphi\) and \(f\) guaranteeing the existence of \(C^2\)-positive solutions of (1). Whenever (1) possesses solutions the existence of a minimal solution \(\widetilde u\) of (1) is proved, which turns out to be the ground state one. Several properties of \(\widetilde u\) such as removability of the possible isolated singularities and its boundary behaviour are investigated in Theorem 1.2 and 1.4.NEWLINENEWLINE The main novelty here is the presence of \(\delta_K(x)\) and its important role in the qualitative study of (1).