Singular solutions of a nonlinear equation in bounded domains of \(\mathbb{R}^2\) (Q697444)

This paper is devoted to the singular solutions of a nonlinear equation in bounded domains of \(\mathbb{R}^2\), that is \[ \begin{cases} \Delta u(x)+f\bigl( x,u(x) \bigr)=0,\quad & x\in\Omega \setminus\{0\},\\ u(x)>0,\quad & x\in \Omega \setminus\{0\},\\ u(x)\sim \log{1\over|x|}, \quad\text{near} & x=0,\\ u(x)=0, \quad & x\in\partial \Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded regular Jordan domain in \(\mathbb{R}^2\) containing 0 and \(f\) is a given measurable function on \(\Omega\times (0,\infty)\). Under suitable assumptions on \(f\), the authors show the existence of infinitely many solutions of (1).