An exact estimate result for a semilinear equation with critical exponent and prescribed singularity (Q335406)

This paper is concerned with the study of positive solutions for the elliptic equation \(-\Delta u-\lambda V(x) u=h(x)u^{-\gamma}+\mu u^{2^8-1}\) in \(\Omega\), subject to \(u=0\) on the boundary \(\partial\Omega\). Here \(\Omega\) is a smooth and bounded domain in \(\mathbb R^N\), \(N\geq 3\), \(0<\gamma<1\), \(2^\ast=\frac{2N}{N-2}\), \(0<\lambda<\bar \lambda=(N-2)^2/4\), \(0<h\in C(\overline{\Omega})\) and \(V\) has prescribed finitely many singular points. The main results of the paper establish the existence of multiple solutions for \(\mu\in (0,\mu^\ast)\) where \(\mu^\ast>0\) is a positive parameter whose estimation is also discussed in the paper.