The existence of positive solutions for a class of indefinite weight semilinear elliptic problems with critical Sobolev exponent (Q1880537)

The author discusses the existence of positive solutions of the following boundary value problems: \[ \begin{cases} -\Delta u=\lambda g(x)f(u) \quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial\Omega, \end{cases}\tag{1} \] where \(\lambda\) is a real parameter, \(\Omega\) is an open bounded domain in \(\mathbb{R}^N\), \(N\geq 3\), with the smooth boundary. The author considers the critical exponent case \(f(u)=u(1+ |u|^p)\) with \(p=\frac{4}{N-2}\). The function \(g,g:\overline\Omega \to\mathbb{R}^1\) is smooth and changes sign.