Existence of positive solution for elliptic equations with singular terms and combined nonlinearities (Q2041710)

This paper studies the following boundary value problem \[ \begin{cases} -\Delta u+u^\gamma=\lambda u^{-q_1}+u^{p_1}\text{ in }\Omega,\\ u>0\text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=\mu u^{-q_2}+u^{p_2}\text{ on }\partial\Omega. \end{cases} \] Here \(0<q_i<1\), \(i=1,2\), \(\lambda\) and \(\mu\) are positive parameters, \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(N\ge 2\), and if \(N\ge 3\), then \(1\leq\gamma<\frac{N+2}{N-2}\), \(1<p_1<\frac{N+2}{N-2}\), \(1<p_2<\frac{N}{N-2}\), and if \(N=2\), then \(1\,\gamma<\infty\), \(1<p_1<\infty\), \(1<p_2<\infty\). The existence and nonexistence of positive solutions for this problem are proved using an approximation scheme, sub-super solution and Perron's method. The asymptotic behavior of the solutions with respect to the parameters is also investigated.