Existence of singular positive solutions of semilinear elliptic equations (Q1299871)

From the introduction. We consider the singular positive solutions of \[ \Delta u+ f(u)= 0\tag{1} \] in \(\mathbb{R}^n\) \((n>2)\), where \(\Delta\) is the \(n\)-dimensional Laplacian. By a singular positive solution we mean a positive classical solution in \(\mathbb{R}^n\setminus\{0\}\) which tends to zero at \(\infty\) and tends to \(\infty\) at the origin. Indeed, it was proved that any positive solution of (1) must be radial at least if \(f\) is assumed to be of class \(C^{1+\varepsilon}\) \((\varepsilon>0)\) on some interval \([0,\delta]\) \((\delta>0)\). Thus we will consider the following problem: \[ u''+ {n-1\over r} u'+ f(u)= 0,\quad u(r)>0,\quad r>0,\tag{2} \] \[ u(r)\to 0,\quad r\to\infty,\quad r=| x|,\quad x\in\mathbb{R}^n.\tag{3} \] In this paper, with a couple of sub- and super-solutions, we prove that problem (2)--(3) possesses infinitely many singular positive solutions by using a monotone iterative method.