Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains (Q5933779)

A semilinear elliptic problem NEWLINE\[NEWLINE \Delta u + \mu g(|x|)f(u) = 0 \;\text{in \(\Omega\)}, \quad u = 0 \;\text{on \(\partial\Omega\)}, \quad \lim_{|x|\to\infty} u(x) = b\geq 0 \tag{1} NEWLINE\]NEWLINE is considered, where \(\Omega\) is the exterior domain \(\{x\in \mathbb R^{n}, |x|> r_0\}\), \(r_0 >0\), \(n\geq 3\), and \(\mu\) is a positive real parameter. Suppose that \(g: \left[r_0, \infty\right[ \to \left]0,\infty\right[\) is continuous, \(\int^\infty_{r_0} rg(r) \roman dr <\infty\), \(f:\mathbb R_{+} \to\mathbb R_{+}\) is continuous, and \(\lim_{u\to\infty} u^{-1} f(u) = \infty\). NEWLINENEWLINENEWLINEIt is proven that if \(b>0\), \(f\) is strictly increasing and \(f(0)=0\) then there exist \(0<\mu_{0} \leq \mu_{f}\) such that the equation (1) has at least two positive radial solutions for \(\mu\in\left]0, \mu_0\right[\), at least one positive radial solution for \(\mu\in \left[\mu_0,\mu_{f}\right]\) and no positive radial solution for \(\mu>\mu_{f}\). The same assertion holds, with \(\mu_0=\mu_{f}\), if \(b=0\), \(f\) is nondecreasing, and \(f(0)>0\).