On a singular semilinear elliptic problem with an asymptotically linear nonlinearity (Q2799610)

In the paper under review, the authors consider the singular semilinear elliptic problem NEWLINE\[NEWLINE\begin{alignedat}{2} -\Delta u & = \lambda f(u)+\mu u^{q-1} \quad & \text{in }& \Omega\, ,\\ u & >0 \quad &\text{in }& \Omega, \\ u & =0 \quad &\text{on }&\partial\Omega,\end{alignedat} NEWLINE\]NEWLINEwhere \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) (\(N \geq 2\)) with smooth boundary, \(0<q<1\), \(f\) is a continuous positive function from \([0,+\infty[\) to \(\mathbb{R}^+\), and \(\lambda\) and \(\mu\) are positive parameters. Then they introduce the associated energy functional: NEWLINE\[NEWLINE \mathcal{E}_{\lambda,\mu}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2-\lambda\int_{\Omega}F(u)-\frac{\mu}{q}\int_{\Omega}(u_+)^q\, . NEWLINE\]NEWLINEUnder the assumption that \(f\) is an asymptotically linear function satisfying a monotonicity condition, they prove existence, uniqueness, and non-existence of the minima of the functional and they obtain a bifurcation type result for the problem.